##### Class 8^{th} Mathematics AP Board Solution

**Exercise 12.1**- 8x, 24 Find the common factors of the given terms in each.
- 3a, 21ab Find the common factors of the given terms in each.
- 7xy, 35x^2 y^3 Find the common factors of the given terms in each.…
- 4m^2 , 6m^2 , 8m^3 Find the common factors of the given terms in each.…
- 15p, 20qr, 25rp Find the common factors of the given terms in each.…
- 4x^2 , 6xy, 8y^2 x Find the common factors of the given terms in each.…
- 12x^2 y, 18xy^2 Find the common factors of the given terms in each.…
- 5x^2 - 25xy Factorise the following expressions
- 9a^2 - 6ax Factorise the following expressions
- 7p^2 + 49pq Factorise the following expressions
- 36a^2 b - 60 a^2 bc Factorise the following expressions
- 3a^2 bc + 6ab^2 c + 9abc^2 Factorise the following expressions
- 4p^2 + 5pq - 6pq^2 Factorise the following expressions
- ut + at^2 Factorise the following expressions
- 3ax - 6xy + 8by - 4ab Factorise the following:
- x^3 + 2x^2 + 5x + 10 Factorise the following:
- m^2 - mn + 4m - 4n Factorise the following:
- a^3 - a^2 b^2 - ab + b^3 Factorise the following:
- p^2 q - pr^2 - pq + r^2 Factorise the following:

**Exercise 12.2**- a^2 + 10a + 25 Factories the following expression-
- l^2 - 16l + 64 Factories the following expression-
- 36x^2 + 96xy + 64y^2 Factories the following expression-
- 25x^2 + 9y^2 - 30xy Factories the following expression-
- 25m^2 - 40mn + 16n^2 Factories the following expression-
- 81x^2 - 198 xy + 121y^2 Factories the following expression-
- (x + y)^2 - 4xy (Hint: first expand (x + y)^2 Factories the following…
- l^4 + 4l^2 m^2 + 4m^4 Factories the following expression-
- x^2 - 36 Factories the following
- 49x^2 - 25y^2 Factories the following
- m^2 - 121 Factories the following
- 81 - 64x^2 Factories the following
- x^2 y^2 - 64 Factories the following
- 6x^2 - 54 Factories the following
- x^2 - 81 Factories the following
- 2x - 32x^5 Factories the following
- 81x^4 - 121x^2 Factories the following
- (p^2 - 2pq + q^2) - r^2 Factories the following
- (x + y)^2 - (x - y)^2 Factories the following
- lx^2 + mx Factories the expressions-
- 7y^2 + 35z^2 Factories the expressions-
- 3x^4 + 6x^3 y + 9x^2 z Factories the expressions-
- x^2 - ax - bx + ab Factories the expressions-
- 3ax - 6ay - 8by + 4bx Factories the expressions-
- mn + m + n + 1 Factories the expressions-
- 6ab - b^2 + 12ac - 2bc Factories the expressions-
- p^2 q - pr^2 - pq + r^2 Factories the expressions-
- x (y + z) - 5 (y + z) Factories the expressions-
- x^4 - y^4 Factories the following
- a^4 - (b + c)^4 Factories the following
- l^2 - (m - n)^2 Factories the following
- 49x^2 - 16/25 Factories the following
- x^4 - 2x^2 y^2 + y^4 Factories the following
- 4 (a + b)^2 - 9 (a - b)^2 Factories the following
- a^2 + 10a + 24 Factories the following expressions
- x^2 + 9x + 18 Factories the following expressions
- p^2 - 10p + 21 Factories the following expressions
- x^2 - 4x - 32 Factories the following expressions
- The lengths of the sides of a triangle are integrals, and its area is also…
- Find the values of ‘m’ for which x^2 + 3xy + x + my -m has two linear factors…

**Exercise 12.3**- 48a^3 by 6a Carry out the following divisions
- 14x^3 by 42x^2 Carry out the following divisions
- 72a^3 b^4 c^5 by 8ab^2 c^3 Carry out the following divisions
- 11xy^2 z^3 by 55xyz Carry out the following divisions
- -54l^4 m^3 n^2 by 9l^2 m^2 n^2 Carry out the following divisions
- (3x^2 - 2x) ÷ x Divide the given polynomial by the given monomial…
- (5a^3 b - 7ab^3) ÷ ab Divide the given polynomial by the given monomial…
- (25x^5 - 15x^4) ÷ 5x^3 Divide the given polynomial by the given monomial…
- 4(l^5 - 6l^4 + 8l^3) ÷ 2l^2 Divide the given polynomial by the given monomial…
- 15 (a^3 b^2 c^2 - a^2 b^3 c^2 + a^2 b^2 c^3) ÷ 3abc Divide the given…
- (3p^3 - 9p^2 q - 6pq^2) ÷ (-3p) Divide the given polynomial by the given…
- (2/3 a^2 b^2 c^2 + 4/3 ab^2 c^2) ÷ 1/2 abc Divide the given polynomial by the…
- (49x - 63) ÷ 7 Workout the following divisions:
- 12x (8x - 20) ÷ 4(2x - 5) Workout the following divisions:
- 11a^3 b^3 (7c - 35) ÷ 3a^2 b^2 (c - 5) Workout the following divisions:…
- 54lmn (l + m) (m + n) (n + l) ÷ 81mn (l + m) (n + l) Workout the following…
- 36 (x + 4) (x^2 + 7x + 10) ÷ 9 (x + 4) Workout the following divisions:…
- a (a + 1) (a + 2) (a + 3) ÷ a (a + 3) Workout the following divisions:…
- (x^2 + 7x + 12) ÷ (x + 3) Factorize the expressions and divide them as…
- (x^2 - 8x + 12) ÷ (x - 6) Factorize the expressions and divide them as…
- (p^2 + 5p + 4) ÷ (p + 1) Factorize the expressions and divide them as…
- 15ab (a^2 -7a + 10) ÷ 3b (a - 2) Factorize the expressions and divide them as…
- 15lm (2p^2 -2q^2) ÷ 3l (p + q) Factorize the expressions and divide them as…
- 26z^3 (32z^2 -18) ÷ 13z^2 (4z - 3) Factorize the expressions and divide them…

**Exercise 12.4**- 3(x - 9) = 3x - 9 Find the errors and correct the following mathematical…
- x(3x + 2) = 3x^2 + 2 Find the errors and correct the following mathematical…
- 2x + 3x = 5x^2 Find the errors and correct the following mathematical…
- 2x + x + 3x = 5x Find the errors and correct the following mathematical…
- 4p + 3p + 2p + p - 9p = 0 Find the errors and correct the following…
- 3x + 2y = 6xy Find the errors and correct the following mathematical sentences…
- (3x)^2 + 4x + 7 = 3x^2 + 4x + 7 Find the errors and correct the following…
- (2x)^2 + 5x = 4x + 5x = 9x Find the errors and correct the following…
- (2a + 3)^2 = 2a^2 + 6a + 9 Find the errors and correct the following…
- Substitute x = - 3 in (a) x^2 + 7x + 12 = (-3)^2 + 7 (-3) + 12 = 9 + 4 + 12 =…
- Substitute x = - 3 in (b) x^2 - 5x + 6 = (-3)^2 -5 (-3) + 6 = 9 - 15 + 6 = 0…
- Substitute x = - 3 in (c) x^2 + 5x = (-3)^2 + 5 (-3) + 6 = - 9 - 15 = -24 Find…
- (x - 4)^2 = x^2 - 16 Find the errors and correct the following mathematical…
- (x + 7)^2 = x^2 + 49 Find the errors and correct the following mathematical…
- (3a + 4b) (a - b) = 3a^2 - 4a^2 Find the errors and correct the following…
- (x + 4) (x + 2) = x^2 + 8 Find the errors and correct the following…
- (x - 4) (x - 2) = x^2 - 8 Find the errors and correct the following…
- 5x^3 ÷ 5x^3 = 0 Find the errors and correct the following mathematical…
- 2x^3 + 1 ÷ 2x^3 = 1 Find the errors and correct the following mathematical…
- 3x + 2 ÷ 3x = 2/3x Find the errors and correct the following mathematical…
- 3x + 5 ÷ 3 = 5 Find the errors and correct the following mathematical…
- 4x+3/3 = x + 1 Find the errors and correct the following mathematical…

**Exercise 12.1**

- 8x, 24 Find the common factors of the given terms in each.
- 3a, 21ab Find the common factors of the given terms in each.
- 7xy, 35x^2 y^3 Find the common factors of the given terms in each.…
- 4m^2 , 6m^2 , 8m^3 Find the common factors of the given terms in each.…
- 15p, 20qr, 25rp Find the common factors of the given terms in each.…
- 4x^2 , 6xy, 8y^2 x Find the common factors of the given terms in each.…
- 12x^2 y, 18xy^2 Find the common factors of the given terms in each.…
- 5x^2 - 25xy Factorise the following expressions
- 9a^2 - 6ax Factorise the following expressions
- 7p^2 + 49pq Factorise the following expressions
- 36a^2 b - 60 a^2 bc Factorise the following expressions
- 3a^2 bc + 6ab^2 c + 9abc^2 Factorise the following expressions
- 4p^2 + 5pq - 6pq^2 Factorise the following expressions
- ut + at^2 Factorise the following expressions
- 3ax - 6xy + 8by - 4ab Factorise the following:
- x^3 + 2x^2 + 5x + 10 Factorise the following:
- m^2 - mn + 4m - 4n Factorise the following:
- a^3 - a^2 b^2 - ab + b^3 Factorise the following:
- p^2 q - pr^2 - pq + r^2 Factorise the following:

**Exercise 12.2**

- a^2 + 10a + 25 Factories the following expression-
- l^2 - 16l + 64 Factories the following expression-
- 36x^2 + 96xy + 64y^2 Factories the following expression-
- 25x^2 + 9y^2 - 30xy Factories the following expression-
- 25m^2 - 40mn + 16n^2 Factories the following expression-
- 81x^2 - 198 xy + 121y^2 Factories the following expression-
- (x + y)^2 - 4xy (Hint: first expand (x + y)^2 Factories the following…
- l^4 + 4l^2 m^2 + 4m^4 Factories the following expression-
- x^2 - 36 Factories the following
- 49x^2 - 25y^2 Factories the following
- m^2 - 121 Factories the following
- 81 - 64x^2 Factories the following
- x^2 y^2 - 64 Factories the following
- 6x^2 - 54 Factories the following
- x^2 - 81 Factories the following
- 2x - 32x^5 Factories the following
- 81x^4 - 121x^2 Factories the following
- (p^2 - 2pq + q^2) - r^2 Factories the following
- (x + y)^2 - (x - y)^2 Factories the following
- lx^2 + mx Factories the expressions-
- 7y^2 + 35z^2 Factories the expressions-
- 3x^4 + 6x^3 y + 9x^2 z Factories the expressions-
- x^2 - ax - bx + ab Factories the expressions-
- 3ax - 6ay - 8by + 4bx Factories the expressions-
- mn + m + n + 1 Factories the expressions-
- 6ab - b^2 + 12ac - 2bc Factories the expressions-
- p^2 q - pr^2 - pq + r^2 Factories the expressions-
- x (y + z) - 5 (y + z) Factories the expressions-
- x^4 - y^4 Factories the following
- a^4 - (b + c)^4 Factories the following
- l^2 - (m - n)^2 Factories the following
- 49x^2 - 16/25 Factories the following
- x^4 - 2x^2 y^2 + y^4 Factories the following
- 4 (a + b)^2 - 9 (a - b)^2 Factories the following
- a^2 + 10a + 24 Factories the following expressions
- x^2 + 9x + 18 Factories the following expressions
- p^2 - 10p + 21 Factories the following expressions
- x^2 - 4x - 32 Factories the following expressions
- The lengths of the sides of a triangle are integrals, and its area is also…
- Find the values of ‘m’ for which x^2 + 3xy + x + my -m has two linear factors…

**Exercise 12.3**

- 48a^3 by 6a Carry out the following divisions
- 14x^3 by 42x^2 Carry out the following divisions
- 72a^3 b^4 c^5 by 8ab^2 c^3 Carry out the following divisions
- 11xy^2 z^3 by 55xyz Carry out the following divisions
- -54l^4 m^3 n^2 by 9l^2 m^2 n^2 Carry out the following divisions
- (3x^2 - 2x) ÷ x Divide the given polynomial by the given monomial…
- (5a^3 b - 7ab^3) ÷ ab Divide the given polynomial by the given monomial…
- (25x^5 - 15x^4) ÷ 5x^3 Divide the given polynomial by the given monomial…
- 4(l^5 - 6l^4 + 8l^3) ÷ 2l^2 Divide the given polynomial by the given monomial…
- 15 (a^3 b^2 c^2 - a^2 b^3 c^2 + a^2 b^2 c^3) ÷ 3abc Divide the given…
- (3p^3 - 9p^2 q - 6pq^2) ÷ (-3p) Divide the given polynomial by the given…
- (2/3 a^2 b^2 c^2 + 4/3 ab^2 c^2) ÷ 1/2 abc Divide the given polynomial by the…
- (49x - 63) ÷ 7 Workout the following divisions:
- 12x (8x - 20) ÷ 4(2x - 5) Workout the following divisions:
- 11a^3 b^3 (7c - 35) ÷ 3a^2 b^2 (c - 5) Workout the following divisions:…
- 54lmn (l + m) (m + n) (n + l) ÷ 81mn (l + m) (n + l) Workout the following…
- 36 (x + 4) (x^2 + 7x + 10) ÷ 9 (x + 4) Workout the following divisions:…
- a (a + 1) (a + 2) (a + 3) ÷ a (a + 3) Workout the following divisions:…
- (x^2 + 7x + 12) ÷ (x + 3) Factorize the expressions and divide them as…
- (x^2 - 8x + 12) ÷ (x - 6) Factorize the expressions and divide them as…
- (p^2 + 5p + 4) ÷ (p + 1) Factorize the expressions and divide them as…
- 15ab (a^2 -7a + 10) ÷ 3b (a - 2) Factorize the expressions and divide them as…
- 15lm (2p^2 -2q^2) ÷ 3l (p + q) Factorize the expressions and divide them as…
- 26z^3 (32z^2 -18) ÷ 13z^2 (4z - 3) Factorize the expressions and divide them…

**Exercise 12.4**

- 3(x - 9) = 3x - 9 Find the errors and correct the following mathematical…
- x(3x + 2) = 3x^2 + 2 Find the errors and correct the following mathematical…
- 2x + 3x = 5x^2 Find the errors and correct the following mathematical…
- 2x + x + 3x = 5x Find the errors and correct the following mathematical…
- 4p + 3p + 2p + p - 9p = 0 Find the errors and correct the following…
- 3x + 2y = 6xy Find the errors and correct the following mathematical sentences…
- (3x)^2 + 4x + 7 = 3x^2 + 4x + 7 Find the errors and correct the following…
- (2x)^2 + 5x = 4x + 5x = 9x Find the errors and correct the following…
- (2a + 3)^2 = 2a^2 + 6a + 9 Find the errors and correct the following…
- Substitute x = - 3 in (a) x^2 + 7x + 12 = (-3)^2 + 7 (-3) + 12 = 9 + 4 + 12 =…
- Substitute x = - 3 in (b) x^2 - 5x + 6 = (-3)^2 -5 (-3) + 6 = 9 - 15 + 6 = 0…
- Substitute x = - 3 in (c) x^2 + 5x = (-3)^2 + 5 (-3) + 6 = - 9 - 15 = -24 Find…
- (x - 4)^2 = x^2 - 16 Find the errors and correct the following mathematical…
- (x + 7)^2 = x^2 + 49 Find the errors and correct the following mathematical…
- (3a + 4b) (a - b) = 3a^2 - 4a^2 Find the errors and correct the following…
- (x + 4) (x + 2) = x^2 + 8 Find the errors and correct the following…
- (x - 4) (x - 2) = x^2 - 8 Find the errors and correct the following…
- 5x^3 ÷ 5x^3 = 0 Find the errors and correct the following mathematical…
- 2x^3 + 1 ÷ 2x^3 = 1 Find the errors and correct the following mathematical…
- 3x + 2 ÷ 3x = 2/3x Find the errors and correct the following mathematical…
- 3x + 5 ÷ 3 = 5 Find the errors and correct the following mathematical…
- 4x+3/3 = x + 1 Find the errors and correct the following mathematical…

###### Exercise 12.1

**Question 1.**Find the common factors of the given terms in each.

8x, 24

**Answer:**∴ Given terms are 8x and 24

Prime factors of given terms are:-

8x = 2 × 2 × 2 × x

24 = 2 × 2 × 2 × 3

As the x is an undefined value,

Common factors will be

2 × 2 × 2 = 8

**Question 2.**Find the common factors of the given terms in each.

3a, 21ab

**Answer:**∴ Given terms are 3a and 21ab

Prime factors of given terms are:-

3a = 3 × a

21ab = a × b × 7 × 3

Common factors will be

⇒ 3 × a = 3a

**Question 3.**Find the common factors of the given terms in each.

7xy, 35x^{2}y^{3}

**Answer:**∴ Given terms are 7xy and 35x^{2}y^{3}

Prime factors of given terms are:-

7xy = 7 × x × y

35x^{2}y^{3} = x × x × y × y × 7 × 5

Common factors will be

⇒ 7 × x × y = 7xy

**Question 4.**Find the common factors of the given terms in each.

4m^{2}, 6m^{2}, 8m^{3}

**Answer:**∴ Given terms are 4m^{2},6m^{2} and 8m^{3}

Prime factors of given terms are:-

4m^{2} = 2 × 2 × m × m

6m^{2} = 3 × 2 × m × m

8m^{3} = 2 × 2 × 2 × m × m × m

Common factors will be

⇒ 2 × m × m = 2m^{2}

**Question 5.**Find the common factors of the given terms in each.

15p, 20qr, 25rp

**Answer:**∴ Given terms are 15p,20qr and 25rp

Prime factors of given terms are:-

15p = 3 × 5 × p

20qr = 2 × 2 × 5 × q × r

25rp = 5 × 5 × r × p

⇒ Common factors will be 5

**Question 6.**Find the common factors of the given terms in each.

4x^{2}, 6xy, 8y^{2}x

**Answer:**∴ Given terms are 4x^{2},6xy and 8y^{2}x

Prime factors of given terms are:-

4x^{2} = 2 × 2 × x × x

6xy = 3 × 2 × x × y

8y^{2}x = 2 × 2 × 2 × y × y × x

Common factors will be

⇒ 2 × x = 2x

**Question 7.**Find the common factors of the given terms in each.

12x^{2}y, 18xy^{2}

**Answer:**Given terms are 12x^{2}yand 18xy^{2}

Prime factors of given terms are:-

12yx^{2} = 3 × 2 × 2 × y × x × x

18xy^{2} = 3 × 2 × 3 × x × y × y

Common factors will be

⇒ 2 × 3 × x × y = 6xy

**Question 8.**Factorise the following expressions

5x^{2} – 25xy

**Answer:**In the given expression

Check the common factors for all terms;

⇒ [5 × x × x - 5 × 5 × x × y]

⇒ 5 × x[x-5 × y]

⇒ 5x[x-5y]

∴ 5x^{2} - 25xy = 5x[x-5y]

**Question 9.**Factorise the following expressions

9a^{2} – 6ax

**Answer:**In the given expression

Check the common factors for all terms;

⇒ [5 × a × a- 2 × 3 × x × a]

⇒ a[5 × a-2 × 3 × x]

⇒ a[5a-6x]

∴ 9a^{2} - 6ax = a[5a-6x]

**Question 10.**Factorise the following expressions

7p^{2} + 49pq

**Answer:**In the given expression

Check the common factors for all terms;

⇒ [7 × p × p + 7 × 7 × p × q]

⇒ 7 × p[p + 7 × q]

⇒ 7p[p + 7q]

∴ 7p^{2} + 49pq = 7p[p + 7q]

**Question 11.**Factorise the following expressions

36a^{2}b – 60 a^{2}bc

**Answer:**In the given expression

Check the common factors for all terms;

⇒ [2 × 2 × 3 × 3 × a × a × b - 2 × 2 × 3 × 5 × a × a × b × c]

⇒ 2 × 2 × 3 × a × a × b[3 × b-5 × c]

⇒ 12a^{2}b[3b-5c]

∴ 36a^{2}b - 60 a^{2}bc = 12a^{2}b[3b-5c]

**Question 12.**Factorise the following expressions

3a^{2}bc + 6ab^{2}c + 9abc^{2}

**Answer:**In the given expression

Check the common factors for all terms;

⇒ [3 × a × a × b × c + 2 × 3 × a × b × b × c + 3 × 3 × a × b × c × c]

⇒ 3 × a × b × c[a + 2 × b + 3 × c]

⇒ 3abc[a + 2b + 3c]

∴ 3a^{2}bc + 6ab^{2}c + 9abc^{2} = 3abc[a + 2b + 3c]

**Question 13.**Factorise the following expressions

4p^{2} + 5pq – 6pq^{2}

**Answer:**In the given expression

Check the common factors for all terms;

⇒ [2 × 2 × p × p + 5 × p × q - 2 × 3 × p × q × q]

⇒ p[2 × 2 × p + 5 × q - 2 × 3 × q × q]

⇒ p[4p + 5q-6q^{2}]

∴ 4p^{2} + 5pq – 6pq^{2} = p[4p + 5q-6q^{2}]

**Question 14.**Factorise the following expressions

ut + at^{2}

**Answer:**In the given expression

Check the common factors for all terms;

⇒ [u × t + a × t × t]

⇒ t[u + a × t]

⇒ t[u + at]

∴ ut + at^{2} = t[u + at]

**Question 15.**Factorise the following:

3ax – 6xy + 8by – 4ab

**Answer:**In the given expression

Check weather there is any common factors for all terms;

None;

Regrouping the 1^{st} 2 terms we have,

3ax-6xy = 3x[a-2y] -------eq 1

Regrouping the last 2 terms we have,

8by-4ab = -4b[a-2y] -------eq 2

∵ we have to make common parts in both eq 1 and 2

Combining eq 1 and 2

3ax – 6xy + 8by – 4ab = 3x[a-2y] + [-4b[a-2y] ]

= 3x[a-2y] - 4b[a-2y]

= [3x-4] [a-2y]

Hence the factors of 3ax – 6xy + 8by – 4ab are [3x-4] and [a-2y]

**Question 16.**Factorise the following:

x^{3} + 2x^{2} + 5x + 10

**Answer:**In the given expression

Check whether there is any common factors for all terms;

None;

Regrouping the 1^{st} 2 terms we have,

x^{3} + 2x^{2} = x^{2}[x + 2] -------eq 1

Regrouping the last 2 terms we have,

5x + 10 = 5[x + 2] -------eq 2

Combining eq 1 and 2

x^{3} + 2x^{2} + 5x + 10 = x^{2}[x + 2] + 5[x + 2]

= [x^{2} + 5][x + 2]

Hence the factors of x^{3} + 2x^{2} + 5x + 10 are [x^{2} + 5] and [x + 2]

**Question 17.**Factorise the following:

m^{2} – mn + 4m – 4n

**Answer:**In the given expression

Check weather there is any common factors for all terms;

None;

Regrouping the 1^{st} 2 terms we have,

m^{2} - mn= m[m - n] -------eq 1

Regrouping the last 2 terms we have,

4m – 4n = 4[m – n] -------eq 2

Combining eq 1 and 2

m^{2} – mn + 4m – 4n = 4[m – n] + m[m - n]

= [4 + m][m-n]

Hence the factors of m^{2} – mn + 4m – 4n are [m – n] and [4 + m]

**Question 18.**Factorise the following:

a^{3} – a^{2}b^{2} – ab + b^{3}

**Answer:**In the given expression

Check weather there is any common factors for all terms;

None;

Regrouping the 1^{st} 2 terms we have,

a^{3} – a^{2}b^{2} = a^{2}[a-b^{2}] -------eq 1

Regrouping the last 2 terms we have,

– ab + b^{3} = -b[a-b^{2}] -------eq 2

∵ we have to make common parts in both eq 1 and 2

Combining eq 1 and 2

a^{3} – a^{2}b^{2} – ab + b^{3} = a^{2}[a-b^{2}] -b[a-b^{2}]

= [a^{2} – b][a – b^{2}]

Hence the factors of a^{3} – a^{2}b^{2} – ab + b^{3} are [a^{2} – b] and[a – b^{2}]

**Question 19.**Factorise the following:

p^{2}q – pr^{2} – pq + r^{2}

**Answer:**In the given expression

Check weather there is any common factors for all terms;

None;

Regrouping the 1^{st} 2 terms we have,

p^{2}q – pr^{2} = p[pq-r^{2}] -------eq 1

Regrouping the last 2 terms we have,

– pq + r^{2} = -1[pq-r^{2}] -------eq 2

∵ we have to make common parts in both eq 1 and 2

Combining eq 1 and 2

p^{2}q – pr^{2} – pq + r^{2} = p[pq-r^{2}] -1[pq-r^{2}]

= [p – 1][pq – r^{2}]

Hence the factors of p^{2}q – pr^{2} – pq + r^{2} are [p – 1] and [pq – r^{2}]

**Question 1.**

Find the common factors of the given terms in each.

8x, 24

**Answer:**

∴ Given terms are 8x and 24

Prime factors of given terms are:-

8x = 2 × 2 × 2 × x

24 = 2 × 2 × 2 × 3

As the x is an undefined value,

Common factors will be

2 × 2 × 2 = 8

**Question 2.**

Find the common factors of the given terms in each.

3a, 21ab

**Answer:**

∴ Given terms are 3a and 21ab

Prime factors of given terms are:-

3a = 3 × a

21ab = a × b × 7 × 3

Common factors will be

⇒ 3 × a = 3a

**Question 3.**

Find the common factors of the given terms in each.

7xy, 35x^{2}y^{3}

**Answer:**

∴ Given terms are 7xy and 35x^{2}y^{3}

Prime factors of given terms are:-

7xy = 7 × x × y

35x^{2}y^{3} = x × x × y × y × 7 × 5

Common factors will be

⇒ 7 × x × y = 7xy

**Question 4.**

Find the common factors of the given terms in each.

4m^{2}, 6m^{2}, 8m^{3}

**Answer:**

∴ Given terms are 4m^{2},6m^{2} and 8m^{3}

Prime factors of given terms are:-

4m^{2} = 2 × 2 × m × m

6m^{2} = 3 × 2 × m × m

8m^{3} = 2 × 2 × 2 × m × m × m

Common factors will be

⇒ 2 × m × m = 2m^{2}

**Question 5.**

Find the common factors of the given terms in each.

15p, 20qr, 25rp

**Answer:**

∴ Given terms are 15p,20qr and 25rp

Prime factors of given terms are:-

15p = 3 × 5 × p

20qr = 2 × 2 × 5 × q × r

25rp = 5 × 5 × r × p

⇒ Common factors will be 5

**Question 6.**

Find the common factors of the given terms in each.

4x^{2}, 6xy, 8y^{2}x

**Answer:**

∴ Given terms are 4x^{2},6xy and 8y^{2}x

Prime factors of given terms are:-

4x^{2} = 2 × 2 × x × x

6xy = 3 × 2 × x × y

8y^{2}x = 2 × 2 × 2 × y × y × x

Common factors will be

⇒ 2 × x = 2x

**Question 7.**

Find the common factors of the given terms in each.

12x^{2}y, 18xy^{2}

**Answer:**

Given terms are 12x^{2}yand 18xy^{2}

Prime factors of given terms are:-

12yx^{2} = 3 × 2 × 2 × y × x × x

18xy^{2} = 3 × 2 × 3 × x × y × y

Common factors will be

⇒ 2 × 3 × x × y = 6xy

**Question 8.**

Factorise the following expressions

5x^{2} – 25xy

**Answer:**

In the given expression

Check the common factors for all terms;

⇒ [5 × x × x - 5 × 5 × x × y]

⇒ 5 × x[x-5 × y]

⇒ 5x[x-5y]

∴ 5x^{2} - 25xy = 5x[x-5y]

**Question 9.**

Factorise the following expressions

9a^{2} – 6ax

**Answer:**

In the given expression

Check the common factors for all terms;

⇒ [5 × a × a- 2 × 3 × x × a]

⇒ a[5 × a-2 × 3 × x]

⇒ a[5a-6x]

∴ 9a^{2} - 6ax = a[5a-6x]

**Question 10.**

Factorise the following expressions

7p^{2} + 49pq

**Answer:**

In the given expression

Check the common factors for all terms;

⇒ [7 × p × p + 7 × 7 × p × q]

⇒ 7 × p[p + 7 × q]

⇒ 7p[p + 7q]

∴ 7p^{2} + 49pq = 7p[p + 7q]

**Question 11.**

Factorise the following expressions

36a^{2}b – 60 a^{2}bc

**Answer:**

In the given expression

Check the common factors for all terms;

⇒ [2 × 2 × 3 × 3 × a × a × b - 2 × 2 × 3 × 5 × a × a × b × c]

⇒ 2 × 2 × 3 × a × a × b[3 × b-5 × c]

⇒ 12a^{2}b[3b-5c]

∴ 36a^{2}b - 60 a^{2}bc = 12a^{2}b[3b-5c]

**Question 12.**

Factorise the following expressions

3a^{2}bc + 6ab^{2}c + 9abc^{2}

**Answer:**

In the given expression

Check the common factors for all terms;

⇒ [3 × a × a × b × c + 2 × 3 × a × b × b × c + 3 × 3 × a × b × c × c]

⇒ 3 × a × b × c[a + 2 × b + 3 × c]

⇒ 3abc[a + 2b + 3c]

∴ 3a^{2}bc + 6ab^{2}c + 9abc^{2} = 3abc[a + 2b + 3c]

**Question 13.**

Factorise the following expressions

4p^{2} + 5pq – 6pq^{2}

**Answer:**

In the given expression

Check the common factors for all terms;

⇒ [2 × 2 × p × p + 5 × p × q - 2 × 3 × p × q × q]

⇒ p[2 × 2 × p + 5 × q - 2 × 3 × q × q]

⇒ p[4p + 5q-6q^{2}]

∴ 4p^{2} + 5pq – 6pq^{2} = p[4p + 5q-6q^{2}]

**Question 14.**

Factorise the following expressions

ut + at^{2}

**Answer:**

In the given expression

Check the common factors for all terms;

⇒ [u × t + a × t × t]

⇒ t[u + a × t]

⇒ t[u + at]

∴ ut + at^{2} = t[u + at]

**Question 15.**

Factorise the following:

3ax – 6xy + 8by – 4ab

**Answer:**

In the given expression

Check weather there is any common factors for all terms;

None;

Regrouping the 1^{st} 2 terms we have,

3ax-6xy = 3x[a-2y] -------eq 1

Regrouping the last 2 terms we have,

8by-4ab = -4b[a-2y] -------eq 2

∵ we have to make common parts in both eq 1 and 2

Combining eq 1 and 2

3ax – 6xy + 8by – 4ab = 3x[a-2y] + [-4b[a-2y] ]

= 3x[a-2y] - 4b[a-2y]

= [3x-4] [a-2y]

Hence the factors of 3ax – 6xy + 8by – 4ab are [3x-4] and [a-2y]

**Question 16.**

Factorise the following:

x^{3} + 2x^{2} + 5x + 10

**Answer:**

In the given expression

Check whether there is any common factors for all terms;

None;

Regrouping the 1^{st} 2 terms we have,

x^{3} + 2x^{2} = x^{2}[x + 2] -------eq 1

Regrouping the last 2 terms we have,

5x + 10 = 5[x + 2] -------eq 2

Combining eq 1 and 2

x^{3} + 2x^{2} + 5x + 10 = x^{2}[x + 2] + 5[x + 2]

= [x^{2} + 5][x + 2]

Hence the factors of x^{3} + 2x^{2} + 5x + 10 are [x^{2} + 5] and [x + 2]

**Question 17.**

Factorise the following:

m^{2} – mn + 4m – 4n

**Answer:**

In the given expression

Check weather there is any common factors for all terms;

None;

Regrouping the 1^{st} 2 terms we have,

m^{2} - mn= m[m - n] -------eq 1

Regrouping the last 2 terms we have,

4m – 4n = 4[m – n] -------eq 2

Combining eq 1 and 2

m^{2} – mn + 4m – 4n = 4[m – n] + m[m - n]

= [4 + m][m-n]

Hence the factors of m^{2} – mn + 4m – 4n are [m – n] and [4 + m]

**Question 18.**

Factorise the following:

a^{3} – a^{2}b^{2} – ab + b^{3}

**Answer:**

In the given expression

Check weather there is any common factors for all terms;

None;

Regrouping the 1^{st} 2 terms we have,

a^{3} – a^{2}b^{2} = a^{2}[a-b^{2}] -------eq 1

Regrouping the last 2 terms we have,

– ab + b^{3} = -b[a-b^{2}] -------eq 2

∵ we have to make common parts in both eq 1 and 2

Combining eq 1 and 2

a^{3} – a^{2}b^{2} – ab + b^{3} = a^{2}[a-b^{2}] -b[a-b^{2}]

= [a^{2} – b][a – b^{2}]

Hence the factors of a^{3} – a^{2}b^{2} – ab + b^{3} are [a^{2} – b] and[a – b^{2}]

**Question 19.**

Factorise the following:

p^{2}q – pr^{2} – pq + r^{2}

**Answer:**

In the given expression

Check weather there is any common factors for all terms;

None;

Regrouping the 1^{st} 2 terms we have,

p^{2}q – pr^{2} = p[pq-r^{2}] -------eq 1

Regrouping the last 2 terms we have,

– pq + r^{2} = -1[pq-r^{2}] -------eq 2

∵ we have to make common parts in both eq 1 and 2

Combining eq 1 and 2

p^{2}q – pr^{2} – pq + r^{2} = p[pq-r^{2}] -1[pq-r^{2}]

= [p – 1][pq – r^{2}]

Hence the factors of p^{2}q – pr^{2} – pq + r^{2} are [p – 1] and [pq – r^{2}]

###### Exercise 12.2

**Question 1.**Factories the following expression-

a^{2} + 10a + 25

**Answer:**In the given expression

1^{st} and last terms are perfect square

⇒ a^{2} = a × a

⇒ 25 = 5 × 5

And the middle expression is in form of 2ab

10a = 2 × 5 × a

∴ a × a + 2 × 5 × a + 5 × 5

Gives (a + b)^{2} = a^{2} + 2ab + b^{2}

⇒ In a^{2} + 10a + 25

a = a and b = 5;

∴ a^{2} + 10a + 25 = (a + 5)^{2}

Hence the factors of a^{2} + 10a + 25 are (a + 5) and (a + 5)

**Question 2.**Factories the following expression-

l^{2} – 16l + 64

**Answer:**In the given expression

1^{st} and last terms are perfect square

⇒ l^{2} = l × l

⇒ 64 = 8 × 8

And the middle expression is in form of 2ab

16l = 2 × 8 × l

∴ l × l + 2 × 8 × l + 8 × 8

Gives (a-b)^{2} = a^{2}-2ab + b^{2}

⇒ In l^{2} + 16l + 64

a = l and b = 8;

∴ l^{2} + 16l + 64 = (l + 8)^{2}

Hence the factors of l^{2} + 16l + 64 are (l + 8) and (l + 8)

**Question 3.**Factories the following expression-

36x^{2} + 96xy + 64y^{2}

**Answer:**In the given expression

1^{st} and last terms are perfect square

⇒ 36x^{2} = 6x × 6x

⇒ 64y^{2} = 8y × 8y

And the middle expression is in form of 2ab

96xy = 2 × 6x × 8y

∴ 6x × 6x + 2 × 8y × 6x + 8y × 8y

Gives (a + b)^{2} = a^{2} + 2ab + b^{2}

⇒ In 36x^{2} + 96xy + 64y^{2}

a = 6x and b = 8y;

∴ 36x^{2} + 96xy + 64y^{2} = (6x + 8y)^{2}

Hence the factors of 36x^{2} + 96xy + 64y^{2} are (6x + 8y) and (6x + 8y)

**Question 4.**Factories the following expression-

25x^{2} + 9y^{2} – 30xy

**Answer:**In the given expression

1^{st} and last terms are perfect square

⇒ 25x^{2} = 5x × 5x

⇒ 9y^{2} = 3y × 3y

And the middle expression is in form of 2ab

30xy = 2 × 5x × 3y

∴ 5x × 5x + 2 × 3y × 5x + 3y × 3y

Gives (a-b)^{2} = a^{2}-2ab + b^{2}

⇒ In 25x^{2} – 30xy + 9y^{2}

a = 5x and b = 3y;

∴ 25x^{2} - 30xy + 9y^{2} = (5x-3y)^{2}

Hence the factors of 25x^{2} - 30xy + 9y^{2} are (5x-3y) and (5x-3y)

**Question 5.**Factories the following expression-

25m^{2} – 40mn + 16n^{2}

**Answer:**In the given expression

1^{st} and last terms are perfect square

⇒ 25m^{2} = 5m × 5m

⇒ 16n^{2} = 4n × 4n

And the middle expression is in form of 2ab

40mn = 2 × 5m × 4n

∴ 5m × 5m - 2 × 4n × 5m + 4n × 4n

Gives (a-b)^{2} = a^{2}-2ab + b^{2}

⇒ In 25m^{2} – 40mn + 16n^{2}

a = 5m and b = 4n;

∴ 25m^{2} – 40mn + 16n^{2} = (5m-4n)^{2}

Hence the factors of 25m^{2} – 40mn + 16n^{2} are (5m-4n)and (5m-4n)

**Question 6.**Factories the following expression-

81x^{2}– 198 xy + 121y^{2}

**Answer:**In the given expression

1^{st} and last terms are perfect square

⇒ 81x^{2} = 9x × 9x

⇒ 121y^{2} = 11y × 11y

And the middle expression is in form of 2ab

198xy = 2 × 9x × 11y

∴ 9x × 9x - 2 × 11y × 9x + 11y × 11y

Gives (a-b)^{2} = a^{2}-2ab + b^{2}

⇒ In 81x^{2} – 198xy + 121y^{2}

a = 9x and b = 11y;

∴ 81x^{2} – 198xy + 121y^{2} = (9x-11y)^{2}

Hence the factors of 81x^{2} – 198xy + 121y^{2} are (9x-11y)and (9x-11y)

**Question 7.**Factories the following expression-

(x + y)^{2} – 4xy

(Hint: first expand (x + y)^{2}

**Answer:**If (a + b)^{2} = a^{2} + 2ab + b^{2}

Then (x + y)^{2} – 4xy

= x^{2} + 2xy + y^{2}-4xy

= x^{2} + y^{2}-2xy

In given expression

1^{st} and last terms are perfect square

⇒ x^{2} = x × x

⇒ y^{2} = y × y

And the middle expression is in form of 2ab

2xy = 2 × x × y

∴ x × x - 2 × y × x + y × y

Gives (a-b)^{2} = a^{2}-2ab + b^{2}

⇒ In x^{2} – 2xy + y^{2}

a = x and b = y;

∴ x^{2} – 2xy + y^{2} = (x-y)^{2}

Hence the factors of (x + y)^{2} – 4xy are (x-y)and (x-y)

**Question 8.**Factories the following expression-

l^{4} + 4l^{2}m^{2} + 4m^{4}

**Answer:**In given expression

1^{st} and last terms are perfect square

⇒ l^{4} = l^{2} × l^{2}

⇒ m^{4} = m^{2} × m^{2}

And the middle expression is in form of 2ab

4l^{2}m^{2} = 2 × l^{2} × m^{2}

∴ l^{2} × l^{2} + 2 × m^{2} × l^{2} + m^{2} × m^{2}

Gives (a + b)^{2} = a^{2} + 2ab + b^{2}

⇒ In l^{4} + 4l^{2}m^{2} + m^{4}

a = l^{2} and b = m^{2};

∴ l^{4} – 4l^{2}m^{2} + m^{4} = (l^{2}-m^{2})^{2}

Hence the factors of l^{4} + 4l^{2}m^{2} + 4m^{4} are (l^{2}-m^{2})and (l^{2}-m^{2})

**Question 9.**Factories the following

x^{2} – 36

**Answer:**In given expression

Both terms are perfect square

⇒ x^{2} = x × x

⇒ 36 = 6 × 6

∴ x^{2}-6^{2}

Seems to be in identity a^{2}-b^{2} = (a + b)(a-b)

Where a = x and b = 6;

x^{2} – 36 = (x + 6)(x-6)

Hence the factors of x^{2} – 36 are (x + 6) and (x-6)

**Question 10.**Factories the following

49x^{2} – 25y^{2}

**Answer:**In given expression

Both terms are perfect square

⇒ 49x^{2} = 7x × 7x

⇒ 25y^{2} = 5y × 5y

∴ 49x^{2}-25y^{2} Seems to be in identity a^{2}-b^{2} = (a + b)(a-b)

Where a = 7x and b = 5y;

49x^{2} – 25y^{2} = (7x + 5y)(7x-5y)

Hence the factors of 49x^{2} – 25y^{2} are (7x + 5y) and (7x-5y)

**Question 11.**Factories the following

m^{2} – 121

**Answer:**In given expression

Both terms are perfect square

⇒ m^{2} = m × m

⇒ 121 = 11 × 11

∴ m^{2}-121 Seems to be in identity a^{2}-b^{2} = (a + b)(a-b)

Where a = m and b = 11;

m^{2} – 121 = (m + 11)(m-11)

Hence the factors of m^{2} – 121 are (m + 11) and (m-11)

**Question 12.**Factories the following

81 – 64x^{2}

**Answer:**In given expression

Both terms are perfect square

⇒ 64x^{2} = 8x × 8x

⇒ 81 = 9 × 9

∴ 81-64x^{2} Seems to be in identity a^{2}-b^{2} = (a + b)(a-b)

Where a = 9 and b = 8x;

81-64x^{2} = (9-8x)(9 + 8x)

Hence the factors of 81-64x^{2}are(9-8x) and (9 + 8x)

**Question 13.**Factories the following

x^{2}y^{2} – 64

**Answer:**In given expression

Both terms are perfect square

⇒ y^{2}x^{2} = xy × xy

⇒ 64 = 8 × 8

∴ x^{2}y^{2} – 64 Seems to be in identity a^{2}-b^{2} = (a + b)(a-b)

Where a = xy and b = 8;

x^{2}y^{2} – 64 = (xy-8)(xy + 8)

Hence the factors of x^{2}y^{2} – 64 are (xy-8) and (xy + 8)

**Question 14.**Factories the following

6x^{2} – 54

**Answer:**In given expression

Take out the common factor,

[2 × 3 × x × x-2 × 3 × 3 × 3]

⇒ 2 × 3[x × x-3 × 3]

⇒ 6[x^{2}-9]

Both terms are perfect square

⇒ x^{2} = x × x

⇒ 9 = 3 × 3

∴ x^{2}– 9 Seems to be in identity a^{2}-b^{2} = (a + b)(a-b)

Where a = x and b = 3;

x^{2}– 9 = (x-3)(x + 3)

Hence the factors of 6x^{2} – 54 are 6,(x-3) and (x + 3)

**Question 15.**Factories the following

x^{2}– 81

**Answer:**In given expression

Both terms are perfect square

⇒ x^{2} = x × x

⇒ 81 = 9 × 9

∴ x^{2} – 81 Seems to be in identity a^{2}-b^{2} = (a + b)(a-b)

Where a = x and b = 9;

x^{2} – 81 = (x-9)(x + 9)

Hence the factors of x^{2} – 81 are (x-9) and (x-9)

**Question 16.**Factories the following

2x – 32x^{5}

**Answer:**In given expression

Take out the common factor,

[2 × x - 2 × 2 × 2 × 2 × 2 × x × x × x × x × x]

⇒ 2 × x[1 - 2 × 2 × 2 × 2 × x × x × x × x]

⇒ 2x [1-16x^{4}] = 2x [1-(2x)^{4}]

⇒ In the term 1-(2x)^{4}

= 1-(4x^{2})^{2}

Both terms are perfect square

⇒ (4x^{2} )^{2} = 4x^{2} × 4x^{2}

⇒ 1 = 1 × 1

∴ 1-(4x^{2})^{2} Seems to be in identity a^{2}-b^{2} = (a + b)(a-b)

Where a = 1 and b = 4x^{2};

1-16x^{4} = (1-4x^{2})(1 + 4x^{2})

→ 1-4x^{2} = 1-(2x)^{2}

∴ 1-4x^{2} Seems to be in identity a^{2}-b^{2} = (a + b)(a-b)

Where a = 1 and b = 2x;

1-4x^{2} = (1-2x)(1 + 2x)

∴ 1-16x^{2} = (1-2x)(1 + 2x) (1 + 4x^{2})

Hence the factors of 2x – 32x^{5} are 2x,(1-2x),(1 + 2x) and (1 + 4x^{2})

**Question 17.**Factories the following

81x^{4} – 121x^{2}

**Answer:**In given expression

Take out the common factor,

[3 × 3 × 3 × 3 × x × x × x × x - 11 × 11 × x × x]

⇒ x × x[3 × 3 × 3 × 3 × x × x - 11 × 11]

⇒ x^{2}[81x^{2} – 121]

In expression 81x^{2} - 121

Both terms are perfect square

⇒ 81x^{2} = 9x × 9x

⇒ 121 = 11 × 11

∴ 81x^{2} – 121 Seems to be in identity a^{2}-b^{2} = (a + b)(a-b)

Where a = 9x and b = 11;

81x^{2} – 121 = (9x-11)(9x + 11)

Hence the factors of 81x^{4} – 121x^{2} are x^{2},(9x-11) and (9x + 11)

**Question 18.**Factories the following

(p^{2} – 2pq + q^{2}) – r^{2}

**Answer:**In the given expression p^{2} – 2pq + q^{2}

1^{st} and last terms are perfect square

⇒ p^{2} = p × p

⇒ q^{2} = q × q

And the middle expression is in form of 2ab

2pq = 2 × p × q

∴ p × p - 2 × p × q + q × q

Gives (a-b)^{2} = a^{2}-2ab + b^{2}

⇒ In p^{2} – 2pq + q^{2}

a = p and b = q;

∴ p^{2} – 2pq + q^{2} = (p-q)^{2}

Now the given expression is (p-q)^{2}– r^{2}

Both terms are perfect square

⇒ (p-q)^{2} = (p-q) × (p-q)

⇒ r^{2} = r × r

∴ (p-q)^{2}– r^{2} Seems to be in identity a^{2}-b^{2} = (a + b)(a-b)

Where a = (p-q) and b = r;

(p-q)^{2}– r^{2} = (p-q-r) (p-q + r)

Hence the factors of (p^{2} – 2pq + q^{2}) – r^{2} are (p-q-r) and (p-q + r)

**Question 19.**Factories the following

(x + y)^{2} – (x – y)^{2}

**Answer:**In the given expression

We know that

(a + b)^{2} = a^{2} + 2ab + b^{2}

(a-b)^{2} = a^{2}-2ab + b^{2}

Hence

If a = x and b = y

(x + y)^{2} – (x – y)^{2} = x^{2} + y^{2} + 2xy – [x^{2} + y^{2}-2xy]

= x^{2} + y^{2} + 2xy -x^{2}-y^{2} + 2xy

= 4xy

**Question 20.**Factories the expressions-

lx^{2} + mx

**Answer:**In the given expression

Take out the common in all the terms,

⇒ lx^{2} + mx

⇒ x[lx + m]

**Question 21.**Factories the expressions-

7y^{2} + 35z^{2}

**Answer:**In the given expression

Take out the common in all the terms,

⇒ 7y^{2} + 35z^{2}

⇒ 7[y^{2} + 5z^{2}]

**Question 22.**Factories the expressions-

3x^{4} + 6x^{3}y + 9x^{2}z

**Answer:**In the given expression

Take out the common in all the terms,

⇒ 3x^{4} + 6x^{3}y + 9x^{2}z

⇒ 3x^{2}[x^{2} + 2xy + 3z]

**Question 23.**Factories the expressions-

x^{2} – ax – bx + ab

**Answer:**In the given expression

Check weather there is any common factors for all terms;

None;

Regrouping the 1^{st} 2 terms we have,

x^{2} - ax= x[x - a] -------eq 1

Regrouping the last 2 terms we have,

-bx + ab = -b[x – a] -------eq 2

Combining eq 1 and 2

x^{2} – ax – bx + ab = x[x - a] - b[x – a]

= [x - b][x - a]

Hence the factors of[ x^{2} – ax – bx + ab] are [x - b]and [x - a]

**Question 24.**Factories the expressions-

3ax – 6ay – 8by + 4bx

**Answer:**In the given expression

Check weather there is any common factors for all terms;

None;

Regrouping the 1^{st} 2 terms we have,

3ax – 6ay= 3a[x - 2y] -------eq 1

Regrouping the last 2 terms we have,

-8by + 4bx = 4b[x – 2y] -------eq 2

Combining eq 1 and 2

3ax – 6ay – 8by + 4bx = 3a[x - 2y] + 4b[x – 2y]

= [x – 2y][3a + 4b]

Hence the factors of[3ax – 6ay – 8by + 4bx] are [x – 2y] and [3a + 4b]

**Question 25.**Factories the expressions-

mn + m + n + 1

**Answer:**In the given expression

Check weather there is any common factors for all terms;

None;

Regrouping the 1^{st} 2 terms we have,

mn + m = m[n + 1] -------eq 1

Regrouping the last 2 terms we have,

n + 1 = 1[n + 1] -------eq 2

Combining eq 1 and 2

mn + m + n + 1 = m[n + 1] + 1[n + 1]

= [m + 1][n + 1]

Hence the factors of[mn + m + n + 1] are [m + 1] and [n + 1]

**Question 26.**Factories the expressions-

6ab – b^{2} + 12ac – 2bc

**Answer:**In the given expression

Check weather there is any common factors for all terms;

None;

Regrouping the 1^{st} 2 terms we have,

6ab – b^{2} = b[6a - b] -------eq 1

Regrouping the last 2 terms we have,

12ac – 2bc = 2c[6a - b] -------eq 2

Combining eq 1 and 2

6ab – b^{2} + 12ac – 2bc = b[6a - b] + 2c[6a - b]

= [6a - b][b + 2c]

Hence the factors of[6ab – b^{2} + 12ac – 2bc] are [6a - b] and[b + 2c]

**Question 27.**Factories the expressions-

p^{2}q – pr^{2} – pq + r^{2}

**Answer:**In the given expression

Check weather there is any common factors for all terms;

None;

Regrouping the 1^{st} 2 terms we have,

p^{2}q – pr^{2} = p[pq – r^{2}] -------eq 1

Regrouping the last 2 terms we have,

– pq + r^{2} = -1[pq – r^{2}] -------eq 2

∵ we have to make common parts in both eq 1 and 2

Combining eq 1 and 2

p^{2}q – pr^{2} – pq + r^{2} = p[pq – r^{2}] -1[pq – r^{2}]

= [pq – r^{2}][p - 1]

Hence the factors of[p^{2}q – pr^{2} – pq + r^{2}] are [pq – r^{2}] and [p - 1]

**Question 28.**Factories the expressions-

x (y + z) – 5 (y + z)

**Answer:**In the given expression

Take out the common in all the terms,

⇒ x (y + z) – 5 (y + z)

⇒ (y + z)(x - 5)

Hence the factors of x (y + z) – 5 (y + z) are (y + z) and (x - 5)

**Question 29.**Factories the following

x^{4} – y^{4}

**Answer:**In expression x^{4} – y^{4}

Both terms are perfect square

⇒ x^{4} = x^{2} × x^{2}

⇒ y^{4} = y^{2} × y^{2}

∴ x^{4} – y^{4} Seems to be in identity a^{2}-b^{2} = (a + b)(a-b)

Where a = x^{2} and b = y^{2};

x^{4} – y^{4} = (x^{2} – y^{2})( x^{2} + y^{2}),

∴ x^{2} – y^{2} Seems to be in identity a^{2}-b^{2} = (a + b)(a-b)

Where a = x and b = y;

x^{2} – y^{2} = (x– y)( x+ y),

⇒ x^{4} – y^{4} = (x– y)( x+ y), ( x^{2} + y^{2})

Hence the factors of x^{4} – y^{4} are (x– y),( x+ y) and ( x^{2} + y^{2})

**Question 30.**Factories the following

a^{4} – (b + c)^{4}

**Answer:**In expression a^{4} – (b + c)^{4}

Both terms are perfect square

⇒ a^{4} = a^{2} × a^{2}

⇒ (b + c)^{4} = (b + c)^{2} × (b + c)^{2}

∴ a^{4} – (b + c)^{4} Seems to be in identity a^{2}-b^{2} = (a + b)(a-b)

Where a = a^{2} and b = (b + c)^{2};

a^{4} – (b + c)^{4} = (a^{2} – (b + c)^{2})( a^{2} + (b + c)^{2}),

∴ a^{2} – (b + c)^{2} Seems to be in identity a^{2}-b^{2} = (a + b)(a-b)

Where a = a and b = (b + c);

a^{2} – (b + c)^{2} = (a– (b + c))( a+ (b + c)),

⇒ a^{4} – (b + c)^{4} = (a– (b + c))( a+ (b + c)), ( a^{2} + (b + c)^{2})

⇒ a^{4} – (b + c)^{4} = (a–b–c)(a + b + c), (a^{2} + b^{2} + c^{2} + 2bc)

Hence the factors of a^{4} – (b + c)^{4} are (a–b–c),(a + b + c),( a^{2} + b^{2} + c^{2} + 2bc)

**Question 31.**Factories the following

l^{2} – (m – n)^{2}

**Answer:**In the given expression l^{2} – (m – n)^{2}

Both terms are perfect square

⇒ l^{2} = l × l

⇒ (m – n)^{2} = (m – n) × (m – n)

∴ l^{2} – (m - n)^{2} Seems to be in identity a^{2}-b^{2} = (a + b)(a-b)

Where a = a and b = (m - n);

∴ l^{2} – (m – n)^{2} = (l + m-n)(l-m + n)

Hence the factors of l^{2}–(m–n)^{2}are (l + m-n)(l-m + n)

**Question 32.**

**Answer:**In the given expression 49x^{2} –

Both terms are perfect square

⇒ 49x^{2} = 7x × 7x

⇒ ()^{2} =

49x^{2} – Seems to be in identity a^{2}-b^{2} = (a + b)(a-b)

Where a = 7x and b = ;

∴ (7x)^{2} –()^{2} = ( 7x – ) (7x + )

Hence the factors of 49x^{2} – are ( 7x – ) and (7x + )

**Question 33.**Factories the following

x^{4} – 2x^{2}y^{2} + y^{4}

**Answer:**In the given expression

1^{st} and last terms are perfect square

⇒ x^{4} = x^{2} × x^{2}

⇒ y^{4} = y^{2} × y^{2}

And the middle expression is in form of 2ab

2x^{2}y^{2} = 2 × x^{2} × y^{2}

∴ x^{2} × x^{2} - 2 × x^{2} × y^{2} + y^{2} × y^{2}

Gives (a-b)^{2} = a^{2}-2ab + b^{2}

⇒ x^{4} – 2x^{2}y^{2} + y^{4}

a = x^{2} and b = y^{2};

∴ x^{4} – 2x^{2}y^{2} + y^{4} = (x^{2} – y^{2}) (x^{2} + y^{2})

Hence the factors of x^{4} – 2x^{2}y^{2} + y^{4} are (x^{2} – y^{2}) and (x^{2} + y^{2})

**Question 34.**Factories the following

4 (a + b)^{2} – 9 (a – b)^{2}

**Answer:**In the given expression

We know that

(a + b)^{2} = a^{2} + 2ab + b^{2}

(a-b)^{2} = a^{2}-2ab + b^{2}

Hence

4[a^{2} + 2ab + b^{2}] – 9[a^{2}-2ab + b^{2}]

4a^{2} + 8ab + 4b^{2} - 9a^{2} + 18ab - 9b^{2}

26ab – 5a^{2} - 5b^{2}

25ab + ab – 5a^{2} – 5b^{2}

[25ab – 5a^{2}] + [ab – 5b^{2}]

5a[5b – a] – b[5b – a]

[5a – b][5b – a]

Hence the factors 4 (a + b)^{2} – 9 (a – b)^{2} are [5a – b] and [5b – a]

**Question 35.**Factories the following expressions

a^{2} + 10a + 24

**Answer:**The given expression looks as

x^{2} + (a + b)x + ab

where a + b = 10; and ab = 24;

factors of 24 their sum

1 × 24 1 + 24 = 25

12 × 2 2 + 12 = 14

6 × 4 6 + 4 = 10

∴ the factors having sum 10 are 6 and 4

a^{2} + 10a + 24 = a^{2} + (6 + 4)a + 24

= a^{2} + 6a + 4a + 24

= a(a + 6) + 4(a + 6)

= (a + 6)(a + 4)

Hence the factors of a^{2} + 10a + 24 are (a + 6) and (a + 4)

**Question 36.**Factories the following expressions

x^{2} + 9x + 18

**Answer:**The given expression looks as

x^{2} + (a + b)x + ab

where a + b = 9; and ab = 18;

factors of 18 their sum

1 × 18 1 + 18 = 19

9 × 2 2 + 9 = 11

6 × 3 6 + 3 = 9

∴ the factors having sum 9 are 6 and 3

x^{2} + 9x + 18 = x^{2} + (6 + 3)x + 18

= x^{2} + 6x + 3x + 18

= x(x + 6) + 3(x + 6)

= (x + 6)(x + 3)

Hence the factors of x^{2} + 9x + 18 are (x + 6) and (x + 3)

**Question 37.**Factories the following expressions

p^{2} – 10p + 21

**Answer:**The given expression looks as

x^{2} + (a + b)x + ab

where a + b = -10; and ab = 21;

factors of 21 their sum

-1 × -21 -1-18 = -19

-7 × -3 -7-3 = -10

∴ the factors having sum -10 are -7 and -3

p^{2} + 9p + 18 = p^{2} + (-7-3)p + 21

= p^{2} -7p-3p + 21

= p(p-7) -3(p-7)

= (p-7)(p-3)

Hence the factors of p^{2} + 9p + 18 are (p-7) and (p-3)

**Question 38.**Factories the following expressions

x^{2} – 4x – 32

**Answer:**The given expression looks as

x^{2} + (a + b)x + ab

where a + b = -4; and ab = -32;

factors of -32 their sum

1 × -32 1-32 = -31

-16 × 2 2 -16 = - 14

-8 × 4 4 -8 = -4

∴ the factors having sum -4 are -8 and 4

x^{2} – 4x – 32 = x^{2} + (4 -8)x - 32

= x^{2} + 4x - 8x - 32

= x(x + 4) -8(x + 4)

= (x + 4)(x-8)

Hence the factors of x^{2} – 4x – 32 are (x + 4) and (x-8)

**Question 39.**The lengths of the sides of a triangle are integrals, and its area is also integer. One side is 21 and the perimeter is 48. Find the shortest side.

**Answer:**A = √ s(s-a)(s-b)(s-c)

If the area is an integer

Then [s(s-a)(s-b)(s-c)] should be proper square

If s = Then s = = 24

Hence ;

A = √ 24(24-a)(24-b)(24-c)

If side of triangle are

a = 21 and b + c = 27

let c be smallest side

then b = 27-c

∴ √ 24(24-21)(24-27 + c)(24-c)

⇒ √ 24 × 3 × (c-3)(24-c)

⇒ √ 2 × 2 × 2 × 3 × 3 × (c-3)(24-c)

⇒ 2 × 3√2(c-3)(24-c)

⇒ 6√2(c-3)(24-c)

∴ the value of [2(c-3)(24-c)] must be a perfect square for area to be a integer

For getting square 2(c-3) should be equal to (24-c)

2(c-3) = (24-c)

2c-6 = 24-c

2c + c = 24 + 6

3c = 10

c = 10; b = 27-c = 27-10 = 17

Hence the size of smallest size is 10.

**Question 40.**Find the values of ‘m’ for which x^{2} + 3xy + x + my –m has two linear factors in x and y, with integer coefficients.

**Answer:**For the given 2 degree equation

That must be equal to(ax + by + c)(dx + e)

= ad.x^{2} + bd.xy + cd.x + ea.x + be.y + ec

= ad.x^{2} + bd.xy + (cd + ea).x + be.y + ec

x^{2} + 3xy + x + my–m = ad.x^{2} + bd.xy + (cd + ea).x + be.y + ec

compare the equation

and take out the coefficient of every term

a.d = 1 ----------1

b.d = 3 ----------2

c.d + e.a = 1 ----------3

b.e = m ----------4

e.c = -m ----------5

⇒ from eq 1; a = d = 1 ∵ all coefficient are integers

After putting result in eq 3; c + e = 1 -------6

After putting result in eq 2; b = 3 --------7

⇒ divide eq 4 and 5

∴ that implies b = -c = -3 ∵ eq 7

Put value of c in eq 6

-3 + e = 1

e = 1 + 3 = 4

Putting value of b and e in eq 4

m = b × e

m = 3 × 4 = 12

**Question 1.**

Factories the following expression-

a^{2} + 10a + 25

**Answer:**

In the given expression

1^{st} and last terms are perfect square

⇒ a^{2} = a × a

⇒ 25 = 5 × 5

And the middle expression is in form of 2ab

10a = 2 × 5 × a

∴ a × a + 2 × 5 × a + 5 × 5

Gives (a + b)^{2} = a^{2} + 2ab + b^{2}

⇒ In a^{2} + 10a + 25

a = a and b = 5;

∴ a^{2} + 10a + 25 = (a + 5)^{2}

Hence the factors of a^{2} + 10a + 25 are (a + 5) and (a + 5)

**Question 2.**

Factories the following expression-

l^{2} – 16l + 64

**Answer:**

In the given expression

1^{st} and last terms are perfect square

⇒ l^{2} = l × l

⇒ 64 = 8 × 8

And the middle expression is in form of 2ab

16l = 2 × 8 × l

∴ l × l + 2 × 8 × l + 8 × 8

Gives (a-b)^{2} = a^{2}-2ab + b^{2}

⇒ In l^{2} + 16l + 64

a = l and b = 8;

∴ l^{2} + 16l + 64 = (l + 8)^{2}

Hence the factors of l^{2} + 16l + 64 are (l + 8) and (l + 8)

**Question 3.**

Factories the following expression-

36x^{2} + 96xy + 64y^{2}

**Answer:**

In the given expression

1^{st} and last terms are perfect square

⇒ 36x^{2} = 6x × 6x

⇒ 64y^{2} = 8y × 8y

And the middle expression is in form of 2ab

96xy = 2 × 6x × 8y

∴ 6x × 6x + 2 × 8y × 6x + 8y × 8y

Gives (a + b)^{2} = a^{2} + 2ab + b^{2}

⇒ In 36x^{2} + 96xy + 64y^{2}

a = 6x and b = 8y;

∴ 36x^{2} + 96xy + 64y^{2} = (6x + 8y)^{2}

Hence the factors of 36x^{2} + 96xy + 64y^{2} are (6x + 8y) and (6x + 8y)

**Question 4.**

Factories the following expression-

25x^{2} + 9y^{2} – 30xy

**Answer:**

In the given expression

1^{st} and last terms are perfect square

⇒ 25x^{2} = 5x × 5x

⇒ 9y^{2} = 3y × 3y

And the middle expression is in form of 2ab

30xy = 2 × 5x × 3y

∴ 5x × 5x + 2 × 3y × 5x + 3y × 3y

Gives (a-b)^{2} = a^{2}-2ab + b^{2}

⇒ In 25x^{2} – 30xy + 9y^{2}

a = 5x and b = 3y;

∴ 25x^{2} - 30xy + 9y^{2} = (5x-3y)^{2}

Hence the factors of 25x^{2} - 30xy + 9y^{2} are (5x-3y) and (5x-3y)

**Question 5.**

Factories the following expression-

25m^{2} – 40mn + 16n^{2}

**Answer:**

In the given expression

1^{st} and last terms are perfect square

⇒ 25m^{2} = 5m × 5m

⇒ 16n^{2} = 4n × 4n

And the middle expression is in form of 2ab

40mn = 2 × 5m × 4n

∴ 5m × 5m - 2 × 4n × 5m + 4n × 4n

Gives (a-b)^{2} = a^{2}-2ab + b^{2}

⇒ In 25m^{2} – 40mn + 16n^{2}

a = 5m and b = 4n;

∴ 25m^{2} – 40mn + 16n^{2} = (5m-4n)^{2}

Hence the factors of 25m^{2} – 40mn + 16n^{2} are (5m-4n)and (5m-4n)

**Question 6.**

Factories the following expression-

81x^{2}– 198 xy + 121y^{2}

**Answer:**

In the given expression

1^{st} and last terms are perfect square

⇒ 81x^{2} = 9x × 9x

⇒ 121y^{2} = 11y × 11y

And the middle expression is in form of 2ab

198xy = 2 × 9x × 11y

∴ 9x × 9x - 2 × 11y × 9x + 11y × 11y

Gives (a-b)^{2} = a^{2}-2ab + b^{2}

⇒ In 81x^{2} – 198xy + 121y^{2}

a = 9x and b = 11y;

∴ 81x^{2} – 198xy + 121y^{2} = (9x-11y)^{2}

Hence the factors of 81x^{2} – 198xy + 121y^{2} are (9x-11y)and (9x-11y)

**Question 7.**

Factories the following expression-

(x + y)^{2} – 4xy

(Hint: first expand (x + y)^{2}

**Answer:**

If (a + b)^{2} = a^{2} + 2ab + b^{2}

Then (x + y)^{2} – 4xy

= x^{2} + 2xy + y^{2}-4xy

= x^{2} + y^{2}-2xy

In given expression

1^{st} and last terms are perfect square

⇒ x^{2} = x × x

⇒ y^{2} = y × y

And the middle expression is in form of 2ab

2xy = 2 × x × y

∴ x × x - 2 × y × x + y × y

Gives (a-b)^{2} = a^{2}-2ab + b^{2}

⇒ In x^{2} – 2xy + y^{2}

a = x and b = y;

∴ x^{2} – 2xy + y^{2} = (x-y)^{2}

Hence the factors of (x + y)^{2} – 4xy are (x-y)and (x-y)

**Question 8.**

Factories the following expression-

l^{4} + 4l^{2}m^{2} + 4m^{4}

**Answer:**

In given expression

1^{st} and last terms are perfect square

⇒ l^{4} = l^{2} × l^{2}

⇒ m^{4} = m^{2} × m^{2}

And the middle expression is in form of 2ab

4l^{2}m^{2} = 2 × l^{2} × m^{2}

∴ l^{2} × l^{2} + 2 × m^{2} × l^{2} + m^{2} × m^{2}

Gives (a + b)^{2} = a^{2} + 2ab + b^{2}

⇒ In l^{4} + 4l^{2}m^{2} + m^{4}

a = l^{2} and b = m^{2};

∴ l^{4} – 4l^{2}m^{2} + m^{4} = (l^{2}-m^{2})^{2}

Hence the factors of l^{4} + 4l^{2}m^{2} + 4m^{4} are (l^{2}-m^{2})and (l^{2}-m^{2})

**Question 9.**

Factories the following

x^{2} – 36

**Answer:**

In given expression

Both terms are perfect square

⇒ x^{2} = x × x

⇒ 36 = 6 × 6

∴ x^{2}-6^{2}

Seems to be in identity a^{2}-b^{2} = (a + b)(a-b)

Where a = x and b = 6;

x^{2} – 36 = (x + 6)(x-6)

Hence the factors of x^{2} – 36 are (x + 6) and (x-6)

**Question 10.**

Factories the following

49x^{2} – 25y^{2}

**Answer:**

In given expression

Both terms are perfect square

⇒ 49x^{2} = 7x × 7x

⇒ 25y^{2} = 5y × 5y

∴ 49x^{2}-25y^{2} Seems to be in identity a^{2}-b^{2} = (a + b)(a-b)

Where a = 7x and b = 5y;

49x^{2} – 25y^{2} = (7x + 5y)(7x-5y)

Hence the factors of 49x^{2} – 25y^{2} are (7x + 5y) and (7x-5y)

**Question 11.**

Factories the following

m^{2} – 121

**Answer:**

In given expression

Both terms are perfect square

⇒ m^{2} = m × m

⇒ 121 = 11 × 11

∴ m^{2}-121 Seems to be in identity a^{2}-b^{2} = (a + b)(a-b)

Where a = m and b = 11;

m^{2} – 121 = (m + 11)(m-11)

Hence the factors of m^{2} – 121 are (m + 11) and (m-11)

**Question 12.**

Factories the following

81 – 64x^{2}

**Answer:**

In given expression

Both terms are perfect square

⇒ 64x^{2} = 8x × 8x

⇒ 81 = 9 × 9

∴ 81-64x^{2} Seems to be in identity a^{2}-b^{2} = (a + b)(a-b)

Where a = 9 and b = 8x;

81-64x^{2} = (9-8x)(9 + 8x)

Hence the factors of 81-64x^{2}are(9-8x) and (9 + 8x)

**Question 13.**

Factories the following

x^{2}y^{2} – 64

**Answer:**

In given expression

Both terms are perfect square

⇒ y^{2}x^{2} = xy × xy

⇒ 64 = 8 × 8

∴ x^{2}y^{2} – 64 Seems to be in identity a^{2}-b^{2} = (a + b)(a-b)

Where a = xy and b = 8;

x^{2}y^{2} – 64 = (xy-8)(xy + 8)

Hence the factors of x^{2}y^{2} – 64 are (xy-8) and (xy + 8)

**Question 14.**

Factories the following

6x^{2} – 54

**Answer:**

In given expression

Take out the common factor,

[2 × 3 × x × x-2 × 3 × 3 × 3]

⇒ 2 × 3[x × x-3 × 3]

⇒ 6[x^{2}-9]

Both terms are perfect square

⇒ x^{2} = x × x

⇒ 9 = 3 × 3

∴ x^{2}– 9 Seems to be in identity a^{2}-b^{2} = (a + b)(a-b)

Where a = x and b = 3;

x^{2}– 9 = (x-3)(x + 3)

Hence the factors of 6x^{2} – 54 are 6,(x-3) and (x + 3)

**Question 15.**

Factories the following

x^{2}– 81

**Answer:**

In given expression

Both terms are perfect square

⇒ x^{2} = x × x

⇒ 81 = 9 × 9

∴ x^{2} – 81 Seems to be in identity a^{2}-b^{2} = (a + b)(a-b)

Where a = x and b = 9;

x^{2} – 81 = (x-9)(x + 9)

Hence the factors of x^{2} – 81 are (x-9) and (x-9)

**Question 16.**

Factories the following

2x – 32x^{5}

**Answer:**

In given expression

Take out the common factor,

[2 × x - 2 × 2 × 2 × 2 × 2 × x × x × x × x × x]

⇒ 2 × x[1 - 2 × 2 × 2 × 2 × x × x × x × x]

⇒ 2x [1-16x^{4}] = 2x [1-(2x)^{4}]

⇒ In the term 1-(2x)^{4}

= 1-(4x^{2})^{2}

Both terms are perfect square

⇒ (4x^{2} )^{2} = 4x^{2} × 4x^{2}

⇒ 1 = 1 × 1

∴ 1-(4x^{2})^{2} Seems to be in identity a^{2}-b^{2} = (a + b)(a-b)

Where a = 1 and b = 4x^{2};

1-16x^{4} = (1-4x^{2})(1 + 4x^{2})

→ 1-4x^{2} = 1-(2x)^{2}

∴ 1-4x^{2} Seems to be in identity a^{2}-b^{2} = (a + b)(a-b)

Where a = 1 and b = 2x;

1-4x^{2} = (1-2x)(1 + 2x)

∴ 1-16x^{2} = (1-2x)(1 + 2x) (1 + 4x^{2})

Hence the factors of 2x – 32x^{5} are 2x,(1-2x),(1 + 2x) and (1 + 4x^{2})

**Question 17.**

Factories the following

81x^{4} – 121x^{2}

**Answer:**

In given expression

Take out the common factor,

[3 × 3 × 3 × 3 × x × x × x × x - 11 × 11 × x × x]

⇒ x × x[3 × 3 × 3 × 3 × x × x - 11 × 11]

⇒ x^{2}[81x^{2} – 121]

In expression 81x^{2} - 121

Both terms are perfect square

⇒ 81x^{2} = 9x × 9x

⇒ 121 = 11 × 11

∴ 81x^{2} – 121 Seems to be in identity a^{2}-b^{2} = (a + b)(a-b)

Where a = 9x and b = 11;

81x^{2} – 121 = (9x-11)(9x + 11)

Hence the factors of 81x^{4} – 121x^{2} are x^{2},(9x-11) and (9x + 11)

**Question 18.**

Factories the following

(p^{2} – 2pq + q^{2}) – r^{2}

**Answer:**

In the given expression p^{2} – 2pq + q^{2}

1^{st} and last terms are perfect square

⇒ p^{2} = p × p

⇒ q^{2} = q × q

And the middle expression is in form of 2ab

2pq = 2 × p × q

∴ p × p - 2 × p × q + q × q

Gives (a-b)^{2} = a^{2}-2ab + b^{2}

⇒ In p^{2} – 2pq + q^{2}

a = p and b = q;

∴ p^{2} – 2pq + q^{2} = (p-q)^{2}

Now the given expression is (p-q)^{2}– r^{2}

Both terms are perfect square

⇒ (p-q)^{2} = (p-q) × (p-q)

⇒ r^{2} = r × r

∴ (p-q)^{2}– r^{2} Seems to be in identity a^{2}-b^{2} = (a + b)(a-b)

Where a = (p-q) and b = r;

(p-q)^{2}– r^{2} = (p-q-r) (p-q + r)

Hence the factors of (p^{2} – 2pq + q^{2}) – r^{2} are (p-q-r) and (p-q + r)

**Question 19.**

Factories the following

(x + y)^{2} – (x – y)^{2}

**Answer:**

In the given expression

We know that

(a + b)^{2} = a^{2} + 2ab + b^{2}

(a-b)^{2} = a^{2}-2ab + b^{2}

Hence

If a = x and b = y

(x + y)^{2} – (x – y)^{2} = x^{2} + y^{2} + 2xy – [x^{2} + y^{2}-2xy]

= x^{2} + y^{2} + 2xy -x^{2}-y^{2} + 2xy

= 4xy

**Question 20.**

Factories the expressions-

lx^{2} + mx

**Answer:**

In the given expression

Take out the common in all the terms,

⇒ lx^{2} + mx

⇒ x[lx + m]

**Question 21.**

Factories the expressions-

7y^{2} + 35z^{2}

**Answer:**

In the given expression

Take out the common in all the terms,

⇒ 7y^{2} + 35z^{2}

⇒ 7[y^{2} + 5z^{2}]

**Question 22.**

Factories the expressions-

3x^{4} + 6x^{3}y + 9x^{2}z

**Answer:**

In the given expression

Take out the common in all the terms,

⇒ 3x^{4} + 6x^{3}y + 9x^{2}z

⇒ 3x^{2}[x^{2} + 2xy + 3z]

**Question 23.**

Factories the expressions-

x^{2} – ax – bx + ab

**Answer:**

In the given expression

Check weather there is any common factors for all terms;

None;

Regrouping the 1^{st} 2 terms we have,

x^{2} - ax= x[x - a] -------eq 1

Regrouping the last 2 terms we have,

-bx + ab = -b[x – a] -------eq 2

Combining eq 1 and 2

x^{2} – ax – bx + ab = x[x - a] - b[x – a]

= [x - b][x - a]

Hence the factors of[ x^{2} – ax – bx + ab] are [x - b]and [x - a]

**Question 24.**

Factories the expressions-

3ax – 6ay – 8by + 4bx

**Answer:**

In the given expression

Check weather there is any common factors for all terms;

None;

Regrouping the 1^{st} 2 terms we have,

3ax – 6ay= 3a[x - 2y] -------eq 1

Regrouping the last 2 terms we have,

-8by + 4bx = 4b[x – 2y] -------eq 2

Combining eq 1 and 2

3ax – 6ay – 8by + 4bx = 3a[x - 2y] + 4b[x – 2y]

= [x – 2y][3a + 4b]

Hence the factors of[3ax – 6ay – 8by + 4bx] are [x – 2y] and [3a + 4b]

**Question 25.**

Factories the expressions-

mn + m + n + 1

**Answer:**

In the given expression

Check weather there is any common factors for all terms;

None;

Regrouping the 1^{st} 2 terms we have,

mn + m = m[n + 1] -------eq 1

Regrouping the last 2 terms we have,

n + 1 = 1[n + 1] -------eq 2

Combining eq 1 and 2

mn + m + n + 1 = m[n + 1] + 1[n + 1]

= [m + 1][n + 1]

Hence the factors of[mn + m + n + 1] are [m + 1] and [n + 1]

**Question 26.**

Factories the expressions-

6ab – b^{2} + 12ac – 2bc

**Answer:**

In the given expression

Check weather there is any common factors for all terms;

None;

Regrouping the 1^{st} 2 terms we have,

6ab – b^{2} = b[6a - b] -------eq 1

Regrouping the last 2 terms we have,

12ac – 2bc = 2c[6a - b] -------eq 2

Combining eq 1 and 2

6ab – b^{2} + 12ac – 2bc = b[6a - b] + 2c[6a - b]

= [6a - b][b + 2c]

Hence the factors of[6ab – b^{2} + 12ac – 2bc] are [6a - b] and[b + 2c]

**Question 27.**

Factories the expressions-

p^{2}q – pr^{2} – pq + r^{2}

**Answer:**

In the given expression

Check weather there is any common factors for all terms;

None;

Regrouping the 1^{st} 2 terms we have,

p^{2}q – pr^{2} = p[pq – r^{2}] -------eq 1

Regrouping the last 2 terms we have,

– pq + r^{2} = -1[pq – r^{2}] -------eq 2

∵ we have to make common parts in both eq 1 and 2

Combining eq 1 and 2

p^{2}q – pr^{2} – pq + r^{2} = p[pq – r^{2}] -1[pq – r^{2}]

= [pq – r^{2}][p - 1]

Hence the factors of[p^{2}q – pr^{2} – pq + r^{2}] are [pq – r^{2}] and [p - 1]

**Question 28.**

Factories the expressions-

x (y + z) – 5 (y + z)

**Answer:**

In the given expression

Take out the common in all the terms,

⇒ x (y + z) – 5 (y + z)

⇒ (y + z)(x - 5)

Hence the factors of x (y + z) – 5 (y + z) are (y + z) and (x - 5)

**Question 29.**

Factories the following

x^{4} – y^{4}

**Answer:**

In expression x^{4} – y^{4}

Both terms are perfect square

⇒ x^{4} = x^{2} × x^{2}

⇒ y^{4} = y^{2} × y^{2}

∴ x^{4} – y^{4} Seems to be in identity a^{2}-b^{2} = (a + b)(a-b)

Where a = x^{2} and b = y^{2};

x^{4} – y^{4} = (x^{2} – y^{2})( x^{2} + y^{2}),

∴ x^{2} – y^{2} Seems to be in identity a^{2}-b^{2} = (a + b)(a-b)

Where a = x and b = y;

x^{2} – y^{2} = (x– y)( x+ y),

⇒ x^{4} – y^{4} = (x– y)( x+ y), ( x^{2} + y^{2})

Hence the factors of x^{4} – y^{4} are (x– y),( x+ y) and ( x^{2} + y^{2})

**Question 30.**

Factories the following

a^{4} – (b + c)^{4}

**Answer:**

In expression a^{4} – (b + c)^{4}

Both terms are perfect square

⇒ a^{4} = a^{2} × a^{2}

⇒ (b + c)^{4} = (b + c)^{2} × (b + c)^{2}

∴ a^{4} – (b + c)^{4} Seems to be in identity a^{2}-b^{2} = (a + b)(a-b)

Where a = a^{2} and b = (b + c)^{2};

a^{4} – (b + c)^{4} = (a^{2} – (b + c)^{2})( a^{2} + (b + c)^{2}),

∴ a^{2} – (b + c)^{2} Seems to be in identity a^{2}-b^{2} = (a + b)(a-b)

Where a = a and b = (b + c);

a^{2} – (b + c)^{2} = (a– (b + c))( a+ (b + c)),

⇒ a^{4} – (b + c)^{4} = (a– (b + c))( a+ (b + c)), ( a^{2} + (b + c)^{2})

⇒ a^{4} – (b + c)^{4} = (a–b–c)(a + b + c), (a^{2} + b^{2} + c^{2} + 2bc)

Hence the factors of a^{4} – (b + c)^{4} are (a–b–c),(a + b + c),( a^{2} + b^{2} + c^{2} + 2bc)

**Question 31.**

Factories the following

l^{2} – (m – n)^{2}

**Answer:**

In the given expression l^{2} – (m – n)^{2}

Both terms are perfect square

⇒ l^{2} = l × l

⇒ (m – n)^{2} = (m – n) × (m – n)

∴ l^{2} – (m - n)^{2} Seems to be in identity a^{2}-b^{2} = (a + b)(a-b)

Where a = a and b = (m - n);

∴ l^{2} – (m – n)^{2} = (l + m-n)(l-m + n)

Hence the factors of l^{2}–(m–n)^{2}are (l + m-n)(l-m + n)

**Question 32.**

**Answer:**

In the given expression 49x^{2} –

Both terms are perfect square

⇒ 49x^{2} = 7x × 7x

⇒ ()^{2} =

49x^{2} – Seems to be in identity a^{2}-b^{2} = (a + b)(a-b)

Where a = 7x and b = ;

∴ (7x)^{2} –()^{2} = ( 7x – ) (7x + )

Hence the factors of 49x^{2} – are ( 7x – ) and (7x + )

**Question 33.**

Factories the following

x^{4} – 2x^{2}y^{2} + y^{4}

**Answer:**

In the given expression

1^{st} and last terms are perfect square

⇒ x^{4} = x^{2} × x^{2}

⇒ y^{4} = y^{2} × y^{2}

And the middle expression is in form of 2ab

2x^{2}y^{2} = 2 × x^{2} × y^{2}

∴ x^{2} × x^{2} - 2 × x^{2} × y^{2} + y^{2} × y^{2}

Gives (a-b)^{2} = a^{2}-2ab + b^{2}

⇒ x^{4} – 2x^{2}y^{2} + y^{4}

a = x^{2} and b = y^{2};

∴ x^{4} – 2x^{2}y^{2} + y^{4} = (x^{2} – y^{2}) (x^{2} + y^{2})

Hence the factors of x^{4} – 2x^{2}y^{2} + y^{4} are (x^{2} – y^{2}) and (x^{2} + y^{2})

**Question 34.**

Factories the following

4 (a + b)^{2} – 9 (a – b)^{2}

**Answer:**

In the given expression

We know that

(a + b)^{2} = a^{2} + 2ab + b^{2}

(a-b)^{2} = a^{2}-2ab + b^{2}

Hence

4[a^{2} + 2ab + b^{2}] – 9[a^{2}-2ab + b^{2}]

4a^{2} + 8ab + 4b^{2} - 9a^{2} + 18ab - 9b^{2}

26ab – 5a^{2} - 5b^{2}

25ab + ab – 5a^{2} – 5b^{2}

[25ab – 5a^{2}] + [ab – 5b^{2}]

5a[5b – a] – b[5b – a]

[5a – b][5b – a]

Hence the factors 4 (a + b)^{2} – 9 (a – b)^{2} are [5a – b] and [5b – a]

**Question 35.**

Factories the following expressions

a^{2} + 10a + 24

**Answer:**

The given expression looks as

x^{2} + (a + b)x + ab

where a + b = 10; and ab = 24;

factors of 24 their sum

1 × 24 1 + 24 = 25

12 × 2 2 + 12 = 14

6 × 4 6 + 4 = 10

∴ the factors having sum 10 are 6 and 4

a^{2} + 10a + 24 = a^{2} + (6 + 4)a + 24

= a^{2} + 6a + 4a + 24

= a(a + 6) + 4(a + 6)

= (a + 6)(a + 4)

Hence the factors of a^{2} + 10a + 24 are (a + 6) and (a + 4)

**Question 36.**

Factories the following expressions

x^{2} + 9x + 18

**Answer:**

The given expression looks as

x^{2} + (a + b)x + ab

where a + b = 9; and ab = 18;

factors of 18 their sum

1 × 18 1 + 18 = 19

9 × 2 2 + 9 = 11

6 × 3 6 + 3 = 9

∴ the factors having sum 9 are 6 and 3

x^{2} + 9x + 18 = x^{2} + (6 + 3)x + 18

= x^{2} + 6x + 3x + 18

= x(x + 6) + 3(x + 6)

= (x + 6)(x + 3)

Hence the factors of x^{2} + 9x + 18 are (x + 6) and (x + 3)

**Question 37.**

Factories the following expressions

p^{2} – 10p + 21

**Answer:**

The given expression looks as

x^{2} + (a + b)x + ab

where a + b = -10; and ab = 21;

factors of 21 their sum

-1 × -21 -1-18 = -19

-7 × -3 -7-3 = -10

∴ the factors having sum -10 are -7 and -3

p^{2} + 9p + 18 = p^{2} + (-7-3)p + 21

= p^{2} -7p-3p + 21

= p(p-7) -3(p-7)

= (p-7)(p-3)

Hence the factors of p^{2} + 9p + 18 are (p-7) and (p-3)

**Question 38.**

Factories the following expressions

x^{2} – 4x – 32

**Answer:**

The given expression looks as

x^{2} + (a + b)x + ab

where a + b = -4; and ab = -32;

factors of -32 their sum

1 × -32 1-32 = -31

-16 × 2 2 -16 = - 14

-8 × 4 4 -8 = -4

∴ the factors having sum -4 are -8 and 4

x^{2} – 4x – 32 = x^{2} + (4 -8)x - 32

= x^{2} + 4x - 8x - 32

= x(x + 4) -8(x + 4)

= (x + 4)(x-8)

Hence the factors of x^{2} – 4x – 32 are (x + 4) and (x-8)

**Question 39.**

The lengths of the sides of a triangle are integrals, and its area is also integer. One side is 21 and the perimeter is 48. Find the shortest side.

**Answer:**

A = √ s(s-a)(s-b)(s-c)

If the area is an integer

Then [s(s-a)(s-b)(s-c)] should be proper square

If s = Then s = = 24

Hence ;

A = √ 24(24-a)(24-b)(24-c)

If side of triangle are

a = 21 and b + c = 27

let c be smallest side

then b = 27-c

∴ √ 24(24-21)(24-27 + c)(24-c)

⇒ √ 24 × 3 × (c-3)(24-c)

⇒ √ 2 × 2 × 2 × 3 × 3 × (c-3)(24-c)

⇒ 2 × 3√2(c-3)(24-c)

⇒ 6√2(c-3)(24-c)

∴ the value of [2(c-3)(24-c)] must be a perfect square for area to be a integer

For getting square 2(c-3) should be equal to (24-c)

2(c-3) = (24-c)

2c-6 = 24-c

2c + c = 24 + 6

3c = 10

c = 10; b = 27-c = 27-10 = 17

Hence the size of smallest size is 10.

**Question 40.**

Find the values of ‘m’ for which x^{2} + 3xy + x + my –m has two linear factors in x and y, with integer coefficients.

**Answer:**

For the given 2 degree equation

That must be equal to(ax + by + c)(dx + e)

= ad.x^{2} + bd.xy + cd.x + ea.x + be.y + ec

= ad.x^{2} + bd.xy + (cd + ea).x + be.y + ec

x^{2} + 3xy + x + my–m = ad.x^{2} + bd.xy + (cd + ea).x + be.y + ec

compare the equation

and take out the coefficient of every term

a.d = 1 ----------1

b.d = 3 ----------2

c.d + e.a = 1 ----------3

b.e = m ----------4

e.c = -m ----------5

⇒ from eq 1; a = d = 1 ∵ all coefficient are integers

After putting result in eq 3; c + e = 1 -------6

After putting result in eq 2; b = 3 --------7

⇒ divide eq 4 and 5

∴ that implies b = -c = -3 ∵ eq 7

Put value of c in eq 6

-3 + e = 1

e = 1 + 3 = 4

Putting value of b and e in eq 4

m = b × e

m = 3 × 4 = 12

###### Exercise 12.3

**Question 1.**Carry out the following divisions

48a^{3} by 6a

**Answer:**In the given term

Dividend = 48a^{3} = 2 × 2 × 2 × 2 × 3 × a × a × a

Divisor = 6a = 2 × 3 × a

= 2 × 2 × 2 × a × a

= 8a^{2}

Hence dividing 48a^{3} by 6a gives 8a^{2}

**Question 2.**Carry out the following divisions

14x^{3} by 42x^{2}

**Answer:**In the given term

Dividend = 14x^{3} = 2 × 7 × x × x × x

Divisor = 42x^{2} = 2 × 3 × 7 × x × x

=

=

Hence dividing 14x^{3} by 42x^{2} gives

**Question 3.**Carry out the following divisions

72a^{3}b^{4}c^{5} by 8ab^{2}c^{3}

**Answer:**In the given term

Dividend = 72a^{3}b^{4}c^{5} = 2 × 2 × 2 × 3 × 3 × a × a × a × b × b × b × b × c × c × c × c × c

Divisor = 8ab^{2}c^{3} = 2 × 2 × 2 × a × b × b × c × c × c

=

= 3 × 3 × a × a × b × b × c × c

= 9a^{2}b^{2}c^{2}

Hence dividing 72a^{3}b^{4}c^{5} by 8ab^{2}c^{3} gives 9a^{2}b^{2}c^{2}

**Question 4.**Carry out the following divisions

11xy^{2}z^{3} by 55xyz

**Answer:**In the given term

Dividend = 11xy^{2}z^{3} = 11 × x × y × y × z × z × z

Divisor = 55xyz = 5 × 11 × x × y × z

=

=

=

Hence dividing 11xy^{2}z^{3} by 55xyz gives

**Question 5.**Carry out the following divisions

–54l^{4}m^{3}n^{2} by 9l^{2}m^{2}n^{2}

**Answer:**In the given term

Dividend = -54l^{4}m^{3}n^{2} = (-1) × 2 × 3 × 3 × 3 × l × l × l × l × m × m × m × n × n

Divisor = 9l^{2}m^{2}n^{2} = 3 × 3 × l × l × m × m × n × n

=

= (-1) × 3 × 2 × l × l × m

= -6l^{2}m

Hence dividing –54l^{4}m^{3}n^{2} by 9l^{2}m^{2}n^{2} gives -6l^{2}m

**Question 6.**Divide the given polynomial by the given monomial

(3x^{2}– 2x) ÷ x

**Answer:**In the given term

Dividend = (3x^{2}– 2x)

Take out the common part in binomial term

= (3 × x × x– 2 × x)

= x(3x-2)

Divisor = x

=

= 3x-2

Hence dividing (3x^{2}– 2x) by x gives out 3x-2

**Question 7.**Divide the given polynomial by the given monomial

(5a^{3}b – 7ab^{3}) ÷ ab

**Answer:**In the given term

Dividend = (5a^{3}b – 7ab^{3})

Take out the common part in binomial term

= (5 × a × a × a × b – 7 × a × b × b × b)

= ab(5a^{2} – 7b^{2})

Divisor = ab

=

= (5a^{2} – 7b^{2})

Hence dividing (5a^{3}b – 7ab^{3}) by ab gives out (5a^{2} – 7b^{2})

**Question 8.**Divide the given polynomial by the given monomial

(25x^{5} – 15x^{4}) ÷ 5x^{3}

**Answer:**In the given term

Dividend = (25x^{5} – 15x^{4})

Take out the common part in binomial term

= (5 × 5 × x × x × x × x × x – 3 × 5 × x × x × x × x)

= (5x – 3)5x^{4}

Divisor = 5x^{3}

=

= (5x – 3)x

= 5x^{2} – 3x

Hence dividing (25x^{5} – 15x^{4}) by 5x^{3} gives out 5x^{2} – 3x

**Question 9.**Divide the given polynomial by the given monomial

4(l^{5} – 6l^{4} + 8l^{3}) ÷ 2l^{2}

**Answer:**In the given term

Dividend = (4l^{5} – 6l^{4} + 8l^{3})

Take out the common part in binomial term

= (2 × 2 × l × l × l × l × l– 3 × 2 × l × l × l × l + 2 × 2 × 2 × l × l × l )

= (2l^{2} – 3l + 4)2l^{3}

Divisor = 2l^{2}

=

= (2l^{2} – 3l + 4)l

= (2l^{3} –2l^{2} + 4l)

Hence dividing 4(l^{5} – 6l^{4} + 8l^{3}) by 2l^{2} gives out (2l^{3} –2l^{2} + 4l)

**Question 10.**Divide the given polynomial by the given monomial

15 (a^{3}b^{2}c^{2}– a^{2}b^{3}c^{2} + a^{2}b^{2}c^{3}) ÷ 3abc

**Answer:**In the given term

Dividend = 15 (a^{3}b^{2}c^{2}– a^{2}b^{3}c^{2} + a^{2}b^{2}c^{3})

Take out the common part in binomial term

= 3 × 5(a × a × a × b × b × c × c– a × a × b × b × b × c × c + a × a × b × b × c × c × c )

= 15 a^{2}b^{2}c^{2}(a – b + c)

Divisor = 3abc

=

= 5abc[a-b + c]

= [5a^{2}bc– 5ab^{2}c + 5abc^{2}]

Hence dividing 15 (a^{3}b^{2}c^{2}– a^{2}b^{3}c^{2} + a^{2}b^{2}c^{3}) by 3abc gives out [5a^{2}bc– 5ab^{2}c + 5abc^{2}]

**Question 11.**Divide the given polynomial by the given monomial

(3p^{3}– 9p^{2}q - 6pq^{2}) ÷ (–3p)

**Answer:**In the given term

Dividend = (3p^{3}– 9p^{2}q - 6pq^{2})

Take out the common part in binomial term

= (3 × p × p × p– 3 × 3 × p × p × q - 2 × 3 × p × q × q )

= 3 × p(p^{2}– 3pq - 2q^{2})

Divisor = (–3p)

=

= (-1) (p^{2}– 3pq - 2q^{2})

= (2q^{2} + 3pq - p^{2})

Hence dividing (3p^{3}– 9p^{2}q - 6pq^{2}) by (–3p) gives out (2q^{2} + 3pq - p^{2})

**Question 12.**Divide the given polynomial by the given monomial

( a^{2}b^{2}c^{2} + ab^{2}c^{2}) ÷abc

**Answer:**In the given term

Dividend = ( a^{2}b^{2}c^{2} + ab^{2}c^{2})

Take out the common part in binomial term

= ( × a × a × b × b × c × c + × 2 × a × b × b × c × c )

= × a × b × b × c × c (a + 2)

= ab _{�}^{2}c^{2}(a + 2)

Divisor = abc

=

=

= bc(a + 2)

Hence dividing ( a^{2}b^{2}c^{2} + ab^{2}c^{2}) by abc gives out bc(a + 2)

**Question 13.**Workout the following divisions:

(49x – 63) ÷ 7

**Answer:**In the given term

Dividend = (49x – 63)

Take out the common part in binomial term

= (7 × 7 × x - 7 × 9)

= 7(7 × x - 9)

= 7(7x - 9)

Divisor = 7

=

= (7x - 9)

Hence dividing (49x – 63) by 7 gives out (7x - 9)

**Question 14.**Workout the following divisions:

12x (8x – 20) ÷ 4(2x – 5)

**Answer:**In the given term

Dividend = 12x (8x – 20)

Take out the common part in binomial term

= 2 × 2 × 3 × x(2 × 2 × 2 × x - 2 × 2 × 5 )

= 2 × 2 × 2 × 2 × 3 × x(2 × x - 5 )

= 48x (2x - 5 )

Divisor = 4(2x – 5)

= 12x

Hence divides 12x (8x – 20) by 4(2x – 5) gives out 12x

**Question 15.**Workout the following divisions:

11a^{3}b^{3}(7c – 35) ÷ 3a^{2}b^{2}(c – 5)

**Answer:**In the given term

Dividend = 11a^{3}b^{3}(7c – 35) ÷ 3a^{2}b^{2}(c – 5)

Take out the common part in binomial term

= 11 × a × a × a × b × b × b (7 × c - 5 × 7 )

= 11 × a × a × a × b × b × b × 7 (c - 5 )

= 77a^{3}b^{3}(c - 5)

Divisor = 3a^{2}b^{2}(c – 5)

=

= ab

Hence dividing 11a^{3}b^{3}(7c – 35) by 3a^{2}b^{2}(c – 5) gives out ab

**Question 16.**Workout the following divisions:

54lmn (l + m) (m + n) (n + l) ÷ 81mn (l + m) (n + l)

**Answer:**In the given term

Dividend = 54lmn (l + m) (m + n) (n + l)

Divisor = 81mn (l + m) (n + l)

=

= l(m + n)

Hence dividing 54lmn (l + m) (m + n) (n + l) by 81mn (l + m)(n + l) gives out l(m + n)

**Question 17.**Workout the following divisions:

36 (x + 4) (x^{2} + 7x + 10) ÷ 9 (x + 4)

**Answer:**In the given term

Dividend = 36 (x + 4) (x^{2} + 7x + 10)

Divisor = 9 (x + 4)

=

= 4(x^{2} + 7x + 10)

= (4x^{2} + 27x + 40)

Hence dividing 36 (x + 4) (x^{2} + 7x + 10) by 9 (x + 4) gives out

(4x^{2} + 27x + 40)

**Question 18.**Workout the following divisions:

a (a + 1) (a + 2) (a + 3) ÷ a (a + 3)

**Answer:**In the given term

Dividend = a (a + 1) (a + 2) (a + 3)

Divisor = a (a + 3)

=

= (a + 1) (a + 2)

Hence dividing a (a + 1) (a + 2) (a + 3) by a (a + 3) gives out

(a + 1) (a + 2)

**Question 19.**Factorize the expressions and divide them as directed:

(x^{2} + 7x + 12) ÷ (x + 3)

**Answer:**In the given term

Dividend = (x^{2} + 7x + 12)

The given expression looks as

x^{2} + (a + b)x + ab

where a + b = 7; and ab = 12;

factors of 12 their sum

1 × 12 1 + 12 = 13

6 × 2 2 + 6 = 8

4 × 3 4 + 3 = 7

∴ the factors having sum 7 are 4 and 3

x^{2} + 7x + 12 = x^{2} + (4 + 3)x + 12

= x^{2} + 4x + 3x + 12

= x(x + 4) + 3(x + 4)

= (x + 4)(x + 3)

Divisor = (x + 3)

=

= (x + 4)

Hence dividing (x^{2} + 7x + 12) by (x + 3) gives out (x + 4)

**Question 20.**Factorize the expressions and divide them as directed:

(x^{2} – 8x + 12) ÷ (x – 6)

**Answer:**In the given term

Dividend = (x^{2} - 8x + 12)

The given expression looks as

x^{2} + (a + b)x + ab

where a + b = -8; and ab = 12;

factors of 12 their sum

-1 × -12 -1-12 = -13

-6 × -2 -2-6 = -8

-4 × -3 -4-3 = -7

∴ the factors having sum 7 are 4 and 3

x^{2} - 8x + 12 = x^{2} + (-6-2)x + 12

= x^{2} - 6x - 2x + 12

= x(x - 6) -2(x - 6)

= (x - 6)(x - 2)

Divisor = (x - 6)

=

= (x – 2)

Hence dividing (x^{2} – 8x + 12) by (x – 6) gives out (x – 2)

**Question 21.**Factorize the expressions and divide them as directed:

(p^{2} + 5p + 4) ÷ (p + 1)

**Answer:**In the given term

Dividend = (p^{2} + 5p + 4)

The given expression looks as

x^{2} + (a + b)x + ab

where a + b = 5; and ab = 4;

factors of 4 their sum

1 × 4 1 + 4 = 5

2 × 2 2 + 2 = 4

∴ the factors having sum 5 are 4 and 1

(p^{2} + 5p + 4) = p^{2} + (4 + 1)p + 4

= p^{2} + 4p + p + 4

= p(p + 4) + 1(p + 4)

= (p + 1)(p + 4)

Divisor = (p + 1)

=

= (p + 4)

Hence dividing (p^{2} + 5p + 4) by (p + 1) gives out (p + 4)

**Question 22.**Factorize the expressions and divide them as directed:

15ab (a^{2}–7a + 10) ÷ 3b (a – 2)

**Answer:**In the given term

Dividend = 15ab (a^{2}–7a + 10)

The given expression (a^{2}–7a + 10) looks as

x^{2} + (a + b) x + ab

where a + b = -7; and ab = 10;

factors of 10 their sum

-1 × -10 -1-10 = -11

-2 × -5 -2-5 = -7

∴ the factors having sum -7 are -2 and -5

(a^{2}–7a + 10) = a^{2} + (-2-5)a + 10

= a^{2}–5a – 2a + 10

= a(a – 5) – 2(a – 5)

= (a – 5)(a – 2)

Divisor = 3b (a – 2)

=

= 5a(a – 5)

Hence dividing 15ab (a^{2}–7a + 10) by 3b (a – 2) gives out 5a(a – 5)

**Question 23.**Factorize the expressions and divide them as directed:

15lm (2p^{2}–2q^{2}) ÷ 3l (p + q)

**Answer:**In the given term

Dividend = 15lm (2p^{2}–2q^{2})

In given expression (2p^{2}–2q^{2})

Take out the common factor in binomial term

⇒ (2 × p × p – 2 × q × q)

→ 2(p^{2} – q^{2})

Both terms are perfect square

⇒ p^{2} = p × p

⇒ q^{2} = q × q

∴ (p^{2} – q^{2}) Seems to be in identity a^{2}-b^{2} = (a + b)(a-b)

Where a = p and b = q;

p^{2} – q^{2} = (p + q)(p – q)

Hence the factors of p^{2} – q^{2} are (p + q) and (p – q)

Divisor = 3l (p + q)

=

=

= 10m(p – q)

Hence dividing 15lm (2p^{2}–2q^{2}) by 3l (p + q) gives out 10m(p – q)

**Question 24.**Factorize the expressions and divide them as directed:

26z^{3}(32z^{2}–18) ÷ 13z^{2}(4z – 3)

**Answer:**In the given term

Dividend = 26z^{3}(32z^{2}–18)

Take out the common factor in binomial term

⇒ 2 × 13 × z × z × z (2 × 2 × 2 × 2 × 2 × z × z – 2 × 3 × 3)

⇒ 2 × 2 × 13 × z × z × z (2 × 2 × 2 × 2 × z × z – 3 × 3)

⇒ 52z^{3}(16z^{2} – 9)

In given expression (16z^{2} – 9)

Both terms are perfect square

⇒ 16z^{2} = 4z × 4z

⇒ 9 = 3 × 3

∴ (16z^{2} – 9) Seems to be in identity a^{2}-b^{2} = (a + b)(a-b)

Where a = 4z and b = 3;

(16z^{2} – 9) = (4z + 3)(4z – 3)

Hence the factors of (16z^{2} – 9)are (4z + 3) and (4z – 3)

Divisor = 13z^{2}(4z – 3)

=

=

= 4z(4z + 3)

Hence dividing 26z^{3}(32z^{2}–18) by 13z^{2}(4z – 3) gives out 4z(4z + 3)

**Question 1.**

Carry out the following divisions

48a^{3} by 6a

**Answer:**

In the given term

Dividend = 48a^{3} = 2 × 2 × 2 × 2 × 3 × a × a × a

Divisor = 6a = 2 × 3 × a

= 2 × 2 × 2 × a × a

= 8a^{2}

Hence dividing 48a^{3} by 6a gives 8a^{2}

**Question 2.**

Carry out the following divisions

14x^{3} by 42x^{2}

**Answer:**

In the given term

Dividend = 14x^{3} = 2 × 7 × x × x × x

Divisor = 42x^{2} = 2 × 3 × 7 × x × x

=

=

Hence dividing 14x^{3} by 42x^{2} gives

**Question 3.**

Carry out the following divisions

72a^{3}b^{4}c^{5} by 8ab^{2}c^{3}

**Answer:**

In the given term

Dividend = 72a^{3}b^{4}c^{5} = 2 × 2 × 2 × 3 × 3 × a × a × a × b × b × b × b × c × c × c × c × c

Divisor = 8ab^{2}c^{3} = 2 × 2 × 2 × a × b × b × c × c × c

=

= 3 × 3 × a × a × b × b × c × c

= 9a^{2}b^{2}c^{2}

Hence dividing 72a^{3}b^{4}c^{5} by 8ab^{2}c^{3} gives 9a^{2}b^{2}c^{2}

**Question 4.**

Carry out the following divisions

11xy^{2}z^{3} by 55xyz

**Answer:**

In the given term

Dividend = 11xy^{2}z^{3} = 11 × x × y × y × z × z × z

Divisor = 55xyz = 5 × 11 × x × y × z

=

=

=

Hence dividing 11xy^{2}z^{3} by 55xyz gives

**Question 5.**

Carry out the following divisions

–54l^{4}m^{3}n^{2} by 9l^{2}m^{2}n^{2}

**Answer:**

In the given term

Dividend = -54l^{4}m^{3}n^{2} = (-1) × 2 × 3 × 3 × 3 × l × l × l × l × m × m × m × n × n

Divisor = 9l^{2}m^{2}n^{2} = 3 × 3 × l × l × m × m × n × n

=

= (-1) × 3 × 2 × l × l × m

= -6l^{2}m

Hence dividing –54l^{4}m^{3}n^{2} by 9l^{2}m^{2}n^{2} gives -6l^{2}m

**Question 6.**

Divide the given polynomial by the given monomial

(3x^{2}– 2x) ÷ x

**Answer:**

In the given term

Dividend = (3x^{2}– 2x)

Take out the common part in binomial term

= (3 × x × x– 2 × x)

= x(3x-2)

Divisor = x

=

= 3x-2

Hence dividing (3x^{2}– 2x) by x gives out 3x-2

**Question 7.**

Divide the given polynomial by the given monomial

(5a^{3}b – 7ab^{3}) ÷ ab

**Answer:**

In the given term

Dividend = (5a^{3}b – 7ab^{3})

Take out the common part in binomial term

= (5 × a × a × a × b – 7 × a × b × b × b)

= ab(5a^{2} – 7b^{2})

Divisor = ab

=

= (5a^{2} – 7b^{2})

Hence dividing (5a^{3}b – 7ab^{3}) by ab gives out (5a^{2} – 7b^{2})

**Question 8.**

Divide the given polynomial by the given monomial

(25x^{5} – 15x^{4}) ÷ 5x^{3}

**Answer:**

In the given term

Dividend = (25x^{5} – 15x^{4})

Take out the common part in binomial term

= (5 × 5 × x × x × x × x × x – 3 × 5 × x × x × x × x)

= (5x – 3)5x^{4}

Divisor = 5x^{3}

=

= (5x – 3)x

= 5x^{2} – 3x

Hence dividing (25x^{5} – 15x^{4}) by 5x^{3} gives out 5x^{2} – 3x

**Question 9.**

Divide the given polynomial by the given monomial

4(l^{5} – 6l^{4} + 8l^{3}) ÷ 2l^{2}

**Answer:**

In the given term

Dividend = (4l^{5} – 6l^{4} + 8l^{3})

Take out the common part in binomial term

= (2 × 2 × l × l × l × l × l– 3 × 2 × l × l × l × l + 2 × 2 × 2 × l × l × l )

= (2l^{2} – 3l + 4)2l^{3}

Divisor = 2l^{2}

=

= (2l^{2} – 3l + 4)l

= (2l^{3} –2l^{2} + 4l)

Hence dividing 4(l^{5} – 6l^{4} + 8l^{3}) by 2l^{2} gives out (2l^{3} –2l^{2} + 4l)

**Question 10.**

Divide the given polynomial by the given monomial

15 (a^{3}b^{2}c^{2}– a^{2}b^{3}c^{2} + a^{2}b^{2}c^{3}) ÷ 3abc

**Answer:**

In the given term

Dividend = 15 (a^{3}b^{2}c^{2}– a^{2}b^{3}c^{2} + a^{2}b^{2}c^{3})

Take out the common part in binomial term

= 3 × 5(a × a × a × b × b × c × c– a × a × b × b × b × c × c + a × a × b × b × c × c × c )

= 15 a^{2}b^{2}c^{2}(a – b + c)

Divisor = 3abc

=

= 5abc[a-b + c]

= [5a^{2}bc– 5ab^{2}c + 5abc^{2}]

Hence dividing 15 (a^{3}b^{2}c^{2}– a^{2}b^{3}c^{2} + a^{2}b^{2}c^{3}) by 3abc gives out [5a^{2}bc– 5ab^{2}c + 5abc^{2}]

**Question 11.**

Divide the given polynomial by the given monomial

(3p^{3}– 9p^{2}q - 6pq^{2}) ÷ (–3p)

**Answer:**

In the given term

Dividend = (3p^{3}– 9p^{2}q - 6pq^{2})

Take out the common part in binomial term

= (3 × p × p × p– 3 × 3 × p × p × q - 2 × 3 × p × q × q )

= 3 × p(p^{2}– 3pq - 2q^{2})

Divisor = (–3p)

=

= (-1) (p^{2}– 3pq - 2q^{2})

= (2q^{2} + 3pq - p^{2})

Hence dividing (3p^{3}– 9p^{2}q - 6pq^{2}) by (–3p) gives out (2q^{2} + 3pq - p^{2})

**Question 12.**

Divide the given polynomial by the given monomial

( a^{2}b^{2}c^{2} + ab^{2}c^{2}) ÷abc

**Answer:**

In the given term

Dividend = ( a^{2}b^{2}c^{2} + ab^{2}c^{2})

Take out the common part in binomial term

= ( × a × a × b × b × c × c + × 2 × a × b × b × c × c )

= × a × b × b × c × c (a + 2)

= ab _{�}^{2}c^{2}(a + 2)

Divisor = abc

=

=

= bc(a + 2)

Hence dividing ( a^{2}b^{2}c^{2} + ab^{2}c^{2}) by abc gives out bc(a + 2)

**Question 13.**

Workout the following divisions:

(49x – 63) ÷ 7

**Answer:**

In the given term

Dividend = (49x – 63)

Take out the common part in binomial term

= (7 × 7 × x - 7 × 9)

= 7(7 × x - 9)

= 7(7x - 9)

Divisor = 7

=

= (7x - 9)

Hence dividing (49x – 63) by 7 gives out (7x - 9)

**Question 14.**

Workout the following divisions:

12x (8x – 20) ÷ 4(2x – 5)

**Answer:**

In the given term

Dividend = 12x (8x – 20)

Take out the common part in binomial term

= 2 × 2 × 3 × x(2 × 2 × 2 × x - 2 × 2 × 5 )

= 2 × 2 × 2 × 2 × 3 × x(2 × x - 5 )

= 48x (2x - 5 )

Divisor = 4(2x – 5)

= 12x

Hence divides 12x (8x – 20) by 4(2x – 5) gives out 12x

**Question 15.**

Workout the following divisions:

11a^{3}b^{3}(7c – 35) ÷ 3a^{2}b^{2}(c – 5)

**Answer:**

In the given term

Dividend = 11a^{3}b^{3}(7c – 35) ÷ 3a^{2}b^{2}(c – 5)

Take out the common part in binomial term

= 11 × a × a × a × b × b × b (7 × c - 5 × 7 )

= 11 × a × a × a × b × b × b × 7 (c - 5 )

= 77a^{3}b^{3}(c - 5)

Divisor = 3a^{2}b^{2}(c – 5)

=

= ab

Hence dividing 11a^{3}b^{3}(7c – 35) by 3a^{2}b^{2}(c – 5) gives out ab

**Question 16.**

Workout the following divisions:

54lmn (l + m) (m + n) (n + l) ÷ 81mn (l + m) (n + l)

**Answer:**

In the given term

Dividend = 54lmn (l + m) (m + n) (n + l)

Divisor = 81mn (l + m) (n + l)

=

= l(m + n)

Hence dividing 54lmn (l + m) (m + n) (n + l) by 81mn (l + m)(n + l) gives out l(m + n)

**Question 17.**

Workout the following divisions:

36 (x + 4) (x^{2} + 7x + 10) ÷ 9 (x + 4)

**Answer:**

In the given term

Dividend = 36 (x + 4) (x^{2} + 7x + 10)

Divisor = 9 (x + 4)

=

= 4(x^{2} + 7x + 10)

= (4x^{2} + 27x + 40)

Hence dividing 36 (x + 4) (x^{2} + 7x + 10) by 9 (x + 4) gives out

(4x^{2} + 27x + 40)

**Question 18.**

Workout the following divisions:

a (a + 1) (a + 2) (a + 3) ÷ a (a + 3)

**Answer:**

In the given term

Dividend = a (a + 1) (a + 2) (a + 3)

Divisor = a (a + 3)

=

= (a + 1) (a + 2)

Hence dividing a (a + 1) (a + 2) (a + 3) by a (a + 3) gives out

(a + 1) (a + 2)

**Question 19.**

Factorize the expressions and divide them as directed:

(x^{2} + 7x + 12) ÷ (x + 3)

**Answer:**

In the given term

Dividend = (x^{2} + 7x + 12)

The given expression looks as

x^{2} + (a + b)x + ab

where a + b = 7; and ab = 12;

factors of 12 their sum

1 × 12 1 + 12 = 13

6 × 2 2 + 6 = 8

4 × 3 4 + 3 = 7

∴ the factors having sum 7 are 4 and 3

x^{2} + 7x + 12 = x^{2} + (4 + 3)x + 12

= x^{2} + 4x + 3x + 12

= x(x + 4) + 3(x + 4)

= (x + 4)(x + 3)

Divisor = (x + 3)

=

= (x + 4)

Hence dividing (x^{2} + 7x + 12) by (x + 3) gives out (x + 4)

**Question 20.**

Factorize the expressions and divide them as directed:

(x^{2} – 8x + 12) ÷ (x – 6)

**Answer:**

In the given term

Dividend = (x^{2} - 8x + 12)

The given expression looks as

x^{2} + (a + b)x + ab

where a + b = -8; and ab = 12;

factors of 12 their sum

-1 × -12 -1-12 = -13

-6 × -2 -2-6 = -8

-4 × -3 -4-3 = -7

∴ the factors having sum 7 are 4 and 3

x^{2} - 8x + 12 = x^{2} + (-6-2)x + 12

= x^{2} - 6x - 2x + 12

= x(x - 6) -2(x - 6)

= (x - 6)(x - 2)

Divisor = (x - 6)

=

= (x – 2)

Hence dividing (x^{2} – 8x + 12) by (x – 6) gives out (x – 2)

**Question 21.**

Factorize the expressions and divide them as directed:

(p^{2} + 5p + 4) ÷ (p + 1)

**Answer:**

In the given term

Dividend = (p^{2} + 5p + 4)

The given expression looks as

x^{2} + (a + b)x + ab

where a + b = 5; and ab = 4;

factors of 4 their sum

1 × 4 1 + 4 = 5

2 × 2 2 + 2 = 4

∴ the factors having sum 5 are 4 and 1

(p^{2} + 5p + 4) = p^{2} + (4 + 1)p + 4

= p^{2} + 4p + p + 4

= p(p + 4) + 1(p + 4)

= (p + 1)(p + 4)

Divisor = (p + 1)

=

= (p + 4)

Hence dividing (p^{2} + 5p + 4) by (p + 1) gives out (p + 4)

**Question 22.**

Factorize the expressions and divide them as directed:

15ab (a^{2}–7a + 10) ÷ 3b (a – 2)

**Answer:**

In the given term

Dividend = 15ab (a^{2}–7a + 10)

The given expression (a^{2}–7a + 10) looks as

x^{2} + (a + b) x + ab

where a + b = -7; and ab = 10;

factors of 10 their sum

-1 × -10 -1-10 = -11

-2 × -5 -2-5 = -7

∴ the factors having sum -7 are -2 and -5

(a^{2}–7a + 10) = a^{2} + (-2-5)a + 10

= a^{2}–5a – 2a + 10

= a(a – 5) – 2(a – 5)

= (a – 5)(a – 2)

Divisor = 3b (a – 2)

=

= 5a(a – 5)

Hence dividing 15ab (a^{2}–7a + 10) by 3b (a – 2) gives out 5a(a – 5)

**Question 23.**

Factorize the expressions and divide them as directed:

15lm (2p^{2}–2q^{2}) ÷ 3l (p + q)

**Answer:**

In the given term

Dividend = 15lm (2p^{2}–2q^{2})

In given expression (2p^{2}–2q^{2})

Take out the common factor in binomial term

⇒ (2 × p × p – 2 × q × q)

→ 2(p^{2} – q^{2})

Both terms are perfect square

⇒ p^{2} = p × p

⇒ q^{2} = q × q

∴ (p^{2} – q^{2}) Seems to be in identity a^{2}-b^{2} = (a + b)(a-b)

Where a = p and b = q;

p^{2} – q^{2} = (p + q)(p – q)

Hence the factors of p^{2} – q^{2} are (p + q) and (p – q)

Divisor = 3l (p + q)

=

=

= 10m(p – q)

Hence dividing 15lm (2p^{2}–2q^{2}) by 3l (p + q) gives out 10m(p – q)

**Question 24.**

Factorize the expressions and divide them as directed:

26z^{3}(32z^{2}–18) ÷ 13z^{2}(4z – 3)

**Answer:**

In the given term

Dividend = 26z^{3}(32z^{2}–18)

Take out the common factor in binomial term

⇒ 2 × 13 × z × z × z (2 × 2 × 2 × 2 × 2 × z × z – 2 × 3 × 3)

⇒ 2 × 2 × 13 × z × z × z (2 × 2 × 2 × 2 × z × z – 3 × 3)

⇒ 52z^{3}(16z^{2} – 9)

In given expression (16z^{2} – 9)

Both terms are perfect square

⇒ 16z^{2} = 4z × 4z

⇒ 9 = 3 × 3

∴ (16z^{2} – 9) Seems to be in identity a^{2}-b^{2} = (a + b)(a-b)

Where a = 4z and b = 3;

(16z^{2} – 9) = (4z + 3)(4z – 3)

Hence the factors of (16z^{2} – 9)are (4z + 3) and (4z – 3)

Divisor = 13z^{2}(4z – 3)

=

=

= 4z(4z + 3)

Hence dividing 26z^{3}(32z^{2}–18) by 13z^{2}(4z – 3) gives out 4z(4z + 3)

###### Exercise 12.4

**Question 1.**Find the errors and correct the following mathematical sentences

3(x – 9) = 3x – 9

**Answer:**If LHS is

3(x – 9)

Then RHS would be

⇒ 3(x – 9)

= 3 × x – 3 × 9

= 3x – 27

The error is 27 instead of 9

Hence 3(x – 9) = 3x – 27

**Question 2.**Find the errors and correct the following mathematical sentences

x(3x + 2) = 3x^{2} + 2

**Answer:**If LHS is

x(3x + 2)

Then RHS would be

⇒ x(3x + 2)

= 3 × x × x – 2 × x

= 3x^{2} – 2x

The error is 2x instead of 2

Hence x(3x + 2) = 3x^{2} + 2x

**Question 3.**Find the errors and correct the following mathematical sentences

2x + 3x = 5x^{2}

**Answer:**If LHS is

2x + 3x

Then RHS would be

⇒ 2x + 3x

= x(2 + 3)

= 5x

The error is 5x instead of 5x^{2}

Hence 2x + 3x = 5x

**Question 4.**Find the errors and correct the following mathematical sentences

2x + x + 3x = 5x

**Answer:**If LHS is

2x + x + 3x = 5x

Then RHS would be

⇒ 2x + x + 3x

= x(2 + 1 + 3)

= 6x

The error is 6x instead of 5x

Hence 2x + x + 3x = 6x

**Question 5.**Find the errors and correct the following mathematical sentences

4p + 3p + 2p + p – 9p = 0

**Answer:**If LHS is

4p + 3p + 2p + p – 9p

Then RHS would be

⇒ 4p + 3p + 2p + p – 9p

= p(4 + 3 + 2 + 1–9)

= p(10–9)

= p

The error is p instead of 0

Hence 4p + 3p + 2p + p–9p = p

**Question 6.**Find the errors and correct the following mathematical sentences

3x + 2y = 6xy

**Answer:**If RHS is

6xy

Then LHS would be

⇒ 6xy

= 2 × 3 × x × y

= 3 × x × 2 × y

= 3x × 2y

The error is sign of multiplication instead of sign of addition

Hence 3x × 2y = 6xy

**Question 7.**Find the errors and correct the following mathematical sentences

(3x)^{2} + 4x + 7 = 3x^{2} + 4x + 7

**Answer:**If LHS is

(3x)^{2} + 4x + 7

Then RHS would be

⇒ (3x)^{2} + 4x + 7

= 3^{2} × x^{2} + 4x + 7

= 9x^{2} + 4x + 7

The error is 9x^{2} instead of 3x^{2}

Hence (3x)^{2} + 4x + 7 = 9x^{2} + 4x + 7

**Question 8.**Find the errors and correct the following mathematical sentences

(2x)^{2} + 5x = 4x + 5x = 9x

**Answer:**If LHS is

(2x)^{2} + 5x

Then RHS would be

⇒ (2x)^{2} + 5x

= 2^{2} × x^{2} + 5x

= 4x^{2} + 5x

The error is 4x^{2} instead of 4x

Hence (2x)^{2} + 5x = 4x^{2} + 5x

**Question 9.**Find the errors and correct the following mathematical sentences

(2a + 3)^{2} = 2a^{2} + 6a + 9

**Answer:**If LHS is

(2a + 3)^{2}

Then RHS would be

⇒ (2a + 3)^{2}

= (2a)^{2} + 3^{2} + 2 × 2a × 3

= 4a^{2} + 9 + 12a

= 4a^{2} + 12a + 9

The error is 4a^{2} instead of 2a^{2} and 12a instead of 6a

Hence = (2a + 3)^{2} = 4a^{2} + 9 + 12a

**Question 10.**Find the errors and correct the following mathematical sentences

Substitute x = – 3 in

(a) x^{2} + 7x + 12 = (–3)^{2} + 7 (–3) + 12 = 9 + 4 + 12 = 25

**Answer:**If LHS is

x^{2} + 7x + 12

Then RHS would be

⇒ x^{2} + 7x + 12

Putting x = (-3)

= (–3)^{2} + 7 (–3) + 12

= 9 + (-21) + 12

= 21-21

= 0

The error is (-21) instead of 4 and end result 0 instead of 25

Hence putting x = (-3) in x^{2} + 7x + 12 results to 0

**Question 11.**Find the errors and correct the following mathematical sentences

Substitute x = – 3 in

(b) x^{2}– 5x + 6 = (–3)^{2} –5 (–3) + 6 = 9 – 15 + 6 = 0

**Answer:**If LHS is

x^{2}– 5x + 6

Then RHS would be

⇒ x^{2}– 5x + 6

Putting x = (-3)

= (–3)^{2} –5 (–3) + 6

= 9 + 15 + 6

= 30

The error is + 15 instead of (-15) and end results to 30 instead of 0

Hence putting x = (-3) in x^{2}– 5x + 6 results to 30

**Question 12.**Find the errors and correct the following mathematical sentences

Substitute x = – 3 in

(c) x^{2} + 5x = (–3)^{2} + 5 (–3) + 6 = – 9 – 15 = –24

**Answer:**If LHS is

x^{2} + 5x

Then RHS would be

⇒ x^{2} + 5x

Putting x = (-3)

= (–3)^{2} + 5 (–3)

= 9 + (-15)

= -6

The error is ( + 9) instead of (-9) and end results to (-6) instead of (-24)

Hence putting x = (-3) in x^{2} + 5x results to (-6)

**Question 13.**Find the errors and correct the following mathematical sentences

(x – 4)^{2} = x^{2} – 16

**Answer:**If LHS is

(x – 4)^{2}

Then RHS would be

⇒ (x – 4)^{2}

= (x)^{2} + 4^{2} – 2 × x × 4

= x^{2} + 16 – 8x

The error is x^{2} + 16 – 8x instead of x^{2} – 16

Hence (x – 4)^{2} = x^{2} + 13 – 8x

**Question 14.**Find the errors and correct the following mathematical sentences

(x + 7)^{2} = x^{2} + 49

**Answer:**If LHS is

(x + 7)^{2}

Then RHS would be

⇒ (x + 7)^{2}

= (x)^{2} + 7^{2} + 2 × x × 7

= x^{2} + 49 + 14

The error is x^{2} + 14x + 49 instead of x^{2} + 49

Hence (x + 7)^{2} = x^{2} + 14x + 49

**Question 15.**Find the errors and correct the following mathematical sentences

(3a + 4b) (a – b) = 3a^{2} – 4a^{2}

**Answer:**For getting in the equation

(a^{2} – b^{2} ) = (a + b)(a-b)

RHS would be

3a^{2} – 4b^{2}

Then LHS would be

⇒ 3a^{2} – 4b^{2}

= (3a – 4b)(3a + 4b)

The error is (a – b) instead of (3a – 4b)

3a^{2} – 4b^{2} instead of 3a^{2} – 4a^{2}

Hence 3a^{2} – 4b^{2} = (3a – 4b)(3a + 4b)

**Question 16.**Find the errors and correct the following mathematical sentences

(x + 4) (x + 2) = x^{2} + 8

**Answer:**If LHS is

(x + 4) (x + 2)

Then RHS would be

⇒ (x + 4) (x + 2)

= x^{2} + 4 × x + 2 × x + 2 × 4

= x^{2} + 4x + 2x + 8

= x^{2} + 6x + 8

The error is x^{2} + 6x + 8 instead of x^{2} + 8

Hence (x + 4) (x + 2) = x^{2} + 6x + 8

**Question 17.**Find the errors and correct the following mathematical sentences

(x – 4) (x – 2) = x^{2} – 8

**Answer:**If LHS is

(x – 4) (x – 2)

Then RHS would be

⇒ (x – 4) (x – 2)

= x^{2} – 4 × x – 2 × x + (-2) × (-4)

= x^{2} – 4x – 2x + 8

= x^{2} – 6x + 8

The error is x^{2} – 6x + 8 instead of x^{2} – 8

Hence (x – 4) (x – 2) = x^{2} – 6x + 8

**Question 18.**Find the errors and correct the following mathematical sentences

5x^{3} ÷ 5x^{3} = 0

**Answer:**If LHS is

5x^{3} ÷ 5x^{3}

Then RHS would be

⇒ 5x^{3} ÷ 5x^{3}

=

= 1

The error is1 instead of 0

Hence 5x^{3} ÷ 5x^{3} = 1

**Question 19.**Find the errors and correct the following mathematical sentences

2x^{3} + 1 ÷ 2x^{3} = 1

**Answer:**If LHS is

(2x^{3} + 1) ÷ 2x^{3}

Then RHS would be

⇒ (2x^{3} + 1) ÷ 2x^{3}

=

=

=

The error is instead of 1

Hence (2x^{3} + 1) ÷ 2x^{3} =

**Question 20.**Find the errors and correct the following mathematical sentences

3x + 2 ÷ 3x =

**Answer:**If LHS is

(3x + 2) ÷ 3x

Then RHS would be

⇒ (3x + 2) ÷ 3x

=

=

=

The error is instead of

Hence (3x + 2 )÷ 3x =

**Question 21.**Find the errors and correct the following mathematical sentences

3x + 5 ÷ 3 = 5

**Answer:**If LHS is

For the complete and perfect division

There must be 3x instead of x

(3x + 5)÷3x

Then RHS would be

⇒ (3x + 5)÷3x

=

=

=

The error is instead of 5 and 3x instead of x

Hence = (3x + 5)÷3x =

**Question 22.**Find the errors and correct the following mathematical sentences

= x + 1

**Answer:**If LHS is

Then RHS would be

⇒

= x +

= x + 1

The error is x + 1 instead of x + 1

Hence x + 1

**Question 1.**

Find the errors and correct the following mathematical sentences

3(x – 9) = 3x – 9

**Answer:**

If LHS is

3(x – 9)

Then RHS would be

⇒ 3(x – 9)

= 3 × x – 3 × 9

= 3x – 27

The error is 27 instead of 9

Hence 3(x – 9) = 3x – 27

**Question 2.**

Find the errors and correct the following mathematical sentences

x(3x + 2) = 3x^{2} + 2

**Answer:**

If LHS is

x(3x + 2)

Then RHS would be

⇒ x(3x + 2)

= 3 × x × x – 2 × x

= 3x^{2} – 2x

The error is 2x instead of 2

Hence x(3x + 2) = 3x^{2} + 2x

**Question 3.**

Find the errors and correct the following mathematical sentences

2x + 3x = 5x^{2}

**Answer:**

If LHS is

2x + 3x

Then RHS would be

⇒ 2x + 3x

= x(2 + 3)

= 5x

The error is 5x instead of 5x^{2}

Hence 2x + 3x = 5x

**Question 4.**

Find the errors and correct the following mathematical sentences

2x + x + 3x = 5x

**Answer:**

If LHS is

2x + x + 3x = 5x

Then RHS would be

⇒ 2x + x + 3x

= x(2 + 1 + 3)

= 6x

The error is 6x instead of 5x

Hence 2x + x + 3x = 6x

**Question 5.**

Find the errors and correct the following mathematical sentences

4p + 3p + 2p + p – 9p = 0

**Answer:**

If LHS is

4p + 3p + 2p + p – 9p

Then RHS would be

⇒ 4p + 3p + 2p + p – 9p

= p(4 + 3 + 2 + 1–9)

= p(10–9)

= p

The error is p instead of 0

Hence 4p + 3p + 2p + p–9p = p

**Question 6.**

Find the errors and correct the following mathematical sentences

3x + 2y = 6xy

**Answer:**

If RHS is

6xy

Then LHS would be

⇒ 6xy

= 2 × 3 × x × y

= 3 × x × 2 × y

= 3x × 2y

The error is sign of multiplication instead of sign of addition

Hence 3x × 2y = 6xy

**Question 7.**

Find the errors and correct the following mathematical sentences

(3x)^{2} + 4x + 7 = 3x^{2} + 4x + 7

**Answer:**

If LHS is

(3x)^{2} + 4x + 7

Then RHS would be

⇒ (3x)^{2} + 4x + 7

= 3^{2} × x^{2} + 4x + 7

= 9x^{2} + 4x + 7

The error is 9x^{2} instead of 3x^{2}

Hence (3x)^{2} + 4x + 7 = 9x^{2} + 4x + 7

**Question 8.**

Find the errors and correct the following mathematical sentences

(2x)^{2} + 5x = 4x + 5x = 9x

**Answer:**

If LHS is

(2x)^{2} + 5x

Then RHS would be

⇒ (2x)^{2} + 5x

= 2^{2} × x^{2} + 5x

= 4x^{2} + 5x

The error is 4x^{2} instead of 4x

Hence (2x)^{2} + 5x = 4x^{2} + 5x

**Question 9.**

Find the errors and correct the following mathematical sentences

(2a + 3)^{2} = 2a^{2} + 6a + 9

**Answer:**

If LHS is

(2a + 3)^{2}

Then RHS would be

⇒ (2a + 3)^{2}

= (2a)^{2} + 3^{2} + 2 × 2a × 3

= 4a^{2} + 9 + 12a

= 4a^{2} + 12a + 9

The error is 4a^{2} instead of 2a^{2} and 12a instead of 6a

Hence = (2a + 3)^{2} = 4a^{2} + 9 + 12a

**Question 10.**

Find the errors and correct the following mathematical sentences

Substitute x = – 3 in

(a) x^{2} + 7x + 12 = (–3)^{2} + 7 (–3) + 12 = 9 + 4 + 12 = 25

**Answer:**

If LHS is

x^{2} + 7x + 12

Then RHS would be

⇒ x^{2} + 7x + 12

Putting x = (-3)

= (–3)^{2} + 7 (–3) + 12

= 9 + (-21) + 12

= 21-21

= 0

The error is (-21) instead of 4 and end result 0 instead of 25

Hence putting x = (-3) in x^{2} + 7x + 12 results to 0

**Question 11.**

Find the errors and correct the following mathematical sentences

Substitute x = – 3 in

(b) x^{2}– 5x + 6 = (–3)^{2} –5 (–3) + 6 = 9 – 15 + 6 = 0

**Answer:**

If LHS is

x^{2}– 5x + 6

Then RHS would be

⇒ x^{2}– 5x + 6

Putting x = (-3)

= (–3)^{2} –5 (–3) + 6

= 9 + 15 + 6

= 30

The error is + 15 instead of (-15) and end results to 30 instead of 0

Hence putting x = (-3) in x^{2}– 5x + 6 results to 30

**Question 12.**

Find the errors and correct the following mathematical sentences

Substitute x = – 3 in

(c) x^{2} + 5x = (–3)^{2} + 5 (–3) + 6 = – 9 – 15 = –24

**Answer:**

If LHS is

x^{2} + 5x

Then RHS would be

⇒ x^{2} + 5x

Putting x = (-3)

= (–3)^{2} + 5 (–3)

= 9 + (-15)

= -6

The error is ( + 9) instead of (-9) and end results to (-6) instead of (-24)

Hence putting x = (-3) in x^{2} + 5x results to (-6)

**Question 13.**

Find the errors and correct the following mathematical sentences

(x – 4)^{2} = x^{2} – 16

**Answer:**

If LHS is

(x – 4)^{2}

Then RHS would be

⇒ (x – 4)^{2}

= (x)^{2} + 4^{2} – 2 × x × 4

= x^{2} + 16 – 8x

The error is x^{2} + 16 – 8x instead of x^{2} – 16

Hence (x – 4)^{2} = x^{2} + 13 – 8x

**Question 14.**

Find the errors and correct the following mathematical sentences

(x + 7)^{2} = x^{2} + 49

**Answer:**

If LHS is

(x + 7)^{2}

Then RHS would be

⇒ (x + 7)^{2}

= (x)^{2} + 7^{2} + 2 × x × 7

= x^{2} + 49 + 14

The error is x^{2} + 14x + 49 instead of x^{2} + 49

Hence (x + 7)^{2} = x^{2} + 14x + 49

**Question 15.**

Find the errors and correct the following mathematical sentences

(3a + 4b) (a – b) = 3a^{2} – 4a^{2}

**Answer:**

For getting in the equation

(a^{2} – b^{2} ) = (a + b)(a-b)

RHS would be

3a^{2} – 4b^{2}

Then LHS would be

⇒ 3a^{2} – 4b^{2}

= (3a – 4b)(3a + 4b)

The error is (a – b) instead of (3a – 4b)

3a^{2} – 4b^{2} instead of 3a^{2} – 4a^{2}

Hence 3a^{2} – 4b^{2} = (3a – 4b)(3a + 4b)

**Question 16.**

Find the errors and correct the following mathematical sentences

(x + 4) (x + 2) = x^{2} + 8

**Answer:**

If LHS is

(x + 4) (x + 2)

Then RHS would be

⇒ (x + 4) (x + 2)

= x^{2} + 4 × x + 2 × x + 2 × 4

= x^{2} + 4x + 2x + 8

= x^{2} + 6x + 8

The error is x^{2} + 6x + 8 instead of x^{2} + 8

Hence (x + 4) (x + 2) = x^{2} + 6x + 8

**Question 17.**

Find the errors and correct the following mathematical sentences

(x – 4) (x – 2) = x^{2} – 8

**Answer:**

If LHS is

(x – 4) (x – 2)

Then RHS would be

⇒ (x – 4) (x – 2)

= x^{2} – 4 × x – 2 × x + (-2) × (-4)

= x^{2} – 4x – 2x + 8

= x^{2} – 6x + 8

The error is x^{2} – 6x + 8 instead of x^{2} – 8

Hence (x – 4) (x – 2) = x^{2} – 6x + 8

**Question 18.**

Find the errors and correct the following mathematical sentences

5x^{3} ÷ 5x^{3} = 0

**Answer:**

If LHS is

5x^{3} ÷ 5x^{3}

Then RHS would be

⇒ 5x^{3} ÷ 5x^{3}

=

= 1

The error is1 instead of 0

Hence 5x^{3} ÷ 5x^{3} = 1

**Question 19.**

Find the errors and correct the following mathematical sentences

2x^{3} + 1 ÷ 2x^{3} = 1

**Answer:**

If LHS is

(2x^{3} + 1) ÷ 2x^{3}

Then RHS would be

⇒ (2x^{3} + 1) ÷ 2x^{3}

=

=

=

The error is instead of 1

Hence (2x^{3} + 1) ÷ 2x^{3} =

**Question 20.**

Find the errors and correct the following mathematical sentences

3x + 2 ÷ 3x =

**Answer:**

If LHS is

(3x + 2) ÷ 3x

Then RHS would be

⇒ (3x + 2) ÷ 3x

=

=

=

The error is instead of

Hence (3x + 2 )÷ 3x =

**Question 21.**

Find the errors and correct the following mathematical sentences

3x + 5 ÷ 3 = 5

**Answer:**

If LHS is

For the complete and perfect division

There must be 3x instead of x

(3x + 5)÷3x

Then RHS would be

⇒ (3x + 5)÷3x

=

=

=

The error is instead of 5 and 3x instead of x

Hence = (3x + 5)÷3x =

**Question 22.**

Find the errors and correct the following mathematical sentences

= x + 1

**Answer:**

If LHS is

Then RHS would be

⇒

= x +

= x + 1

The error is x + 1 instead of x + 1

Hence x + 1