# Factorisation Of Algebraic Expressions

##### Class 8th Mathematics (new) MHB Solution

##### Class 8^{th} Mathematics (new) MHB Solution

**Practice Set 6.1**

- Factorise.x2 + 9x + 18
- Factorise.x2 – 10x + 9
- Factorise.y2 + 24y + 144
- Factorise.5y2 + 5y – 10
- Factorise.p2 – 2p – 35
- Factorise.p2 – 7p – 44
- Factorise.m2 – 23m + 120
- Factorise.m2 – 25m + 100
- Factorise.3x2 + 14x + 15
- Factorise.2x2 + x – 45
- Factorise.20x2 – 26x + 8
- Factorise.44x2 – x – 3

**Practice Set 6.2**

- Factorise.x3 + 64y3
- Factorise.125p3 + q3
- Factorise.125k3 + 27m3
- Factorise.2l3 + 432m3
- Factorise.24a3 + 81b3
- Factorise. y^{3} + {1}/{ 8y^{3} }
- Factorise. a^{3} + {8}/{ a^{3} }
- Factorise. 1 + { q^{3} }/{125}

**Practice Set 6.3**

- Factorise :y3 – 27
- Factorise :x3 – 64y3
- Factorise :27m3 – 216n3
- Factorise :125y3 – 1
- Factorise :8p3 – 27/p3
- Factorise :343a3 – 512b3
- Factorise :64x2 – 729y2
- Factorise :16 a3 – 128/b3
- Simplify :(x + y)3 – (x – y)3
- Simplify :(3a + 5b)3 – (3a – 5b)3
- Simplify :(a + b)3 – a3 – b3
- Simplify :p3 – (p + 1)3
- Simplify :(3xy – 2ab)3 – (3xy + 2ab)3

**Practice Set 6.4**

- Simplify: { m^{2} - n^{2} }/{ (m+11) } x frac { m^{2} + mn+n^{2} }/{ m^{3} - n^{3}…
- Simplify: { a^{2} + 10a+21 }/{ a^{2} + 6a-7 } x frac { a^{2} - 1 }/{a+3}…
- Simplify: { 8x^{3} - 27y^{3} }/{ 4x^{2} - 9y^{2} }
- Simplify: { x^{2} - 5x-24 }/{ (x+3) (x+8) } x frac { x^{2} - 64 }/{ (x-8)^{2} }…
- Simplify: { 3x^{2} - x-2 }/{ x^{2} - 7x+12 } / frac { 3x^{2} - 7x-6 }/{ x^{2} -…
- Simplify: { 4x^{2} - 11x+6 }/{ 16x^{2} - 9 }
- Simplify: { a^{3} - 27 }/{ 5a^{2} - 16a+3 } / frac { a^{2} + 3a+9 }/{ 25a^{2} -…
- Simplify: { 1-2x+x^{2} }/{ 1-x^{3} } x frac { 1+x+x^{2} }/{1+x}…

###### Practice Set 6.1

**Question 1.**

Factorise.

x^{2} + 9x + 18

**Answer:**

On comparing with standard quadratic equation that is

we have,

a = 1, b = 9 and c = 18

Now here,

Product ac = 118 = 18

Factors of 18; 29 and 63

Sum should be b = + 9

From above factors (+ 6x + 3x)

will give + 9x sum

therefore + 9x is replaced by ( + 6x + 3x)

Now above eq. becomes

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

⇒ ; taking x common

⇒ (x + 3)(x + 6)

**Question 2.**

Factorise.

x^{2} – 10x + 9

**Answer:**

On comparing with standard quadratic equation that is

we have,0

a = 1, b = – 10 and c = 9

Now here,

Product ac = 19 = 9

Factors of 9; 19 and 33

Sum should be b = – 10

From above factors ( – 1x – 9x)

will give – 10x sum

therefore – 10x is replaced by ( – 1x – 9x)

Now above eq. becomes

; taking x and – 9 common

(x – 1)(x – 9)

**Question 3.**

Factorise.

y^{2} + 24y + 144

**Answer:**

On comparing with standard quadratic equation that is

we have,

a = 1, b = + 24 and c = + 144

Now here,

Product a × c = 1 × 144 = 144

Factors of 144; 12 × 12; 24 × 6;144 × 1;

48 × 3; 72 × 2

Sum should be b = 24

From above factors (12y + 12y)

will give + 24y sum

therefore + 24 is replaced by ( + 12y + 12y)

Now above eq. becomes

; taking y and + 12 common

(y + 1)(y + 12)

Note: Try to find all factors of “c”, then choose from it that combination whose sum or difference give “b”

**Question 4.**

Factorise.

5y^{2} + 5y – 10

**Answer:**

On comparing with standard quadratic equation that is

we have,

a = 5, b = + 5 and c = – 10

Now here,

Product a × c = 5 × – 10 = – 50

Factors of 50; 5 × 10; 25 × 2;50 × 1

Sum should be b = + 5

From above factors ( – 5y + 10y)

will give + 5y sum

therefore + 5y is replaced by ( – 5y + 10y)

Now above eq. becomes

; taking 5y and + 10 common

(y – 1)(5y + 10)

5(y – 1)(y + 2); 5 common

Note: if given equation’s constant a,b,c have common multiple take it out and then factorize.

**Question 5.**

Factorise.

p^{2} – 2p – 35

**Answer:**

On comparing with standard quadratic equation that is

we have,

a = 1, b = – 2 and c = – 35

Now here,

Product a × c = 1 × – 35 = – 35

Factors of 35; 1 × 35 and 7 × 5

Sum should be b = – 2

From above factors ( – 7p + 5p)

will give – 2p sum

therefore – 2p is replaced by ( – 7p + 5p)

Now above eq. becomes

; taking p and + 5 common

(p – 7)(p + 5)

**Question 6.**

Factorise.

p^{2} – 7p – 44

**Answer:**

On comparing with standard quadratic equation that is

we have,

a = 1, b = – 7 and c = – 44

Now here,

Product a × c = 1 × – 44 = – 44

Factors of 44; 1 × 44; 2 × 22;4 × 11

Sum should be b = – 7

From above factors ( – 11p + 4p)

will give – 7p sum

therefore – 7p is replaced by ( – 11p + 4p)

Now above eq. becomes

; taking p and + 4 common

(p + 4)(p – 11)

**Question 7.**

Factorise.

m^{2} – 23m + 120

**Answer:**

On comparing with standard quadratic equation that is

a

we have,

a = 1, b = – 23 and c = + 120

Now here,

Product a × c = 1 × + 120 = + 120

Factors of + 120; 1 × 120; 2 × 60; 4 × 30; 8 × 15; 24 × 5; 40 × 3

Sum should be b = – 23

From above factors ( – 15m – 8m)

will give – 23m sum

therefore – 23m is replaced by ( – 15m – 8m)

Now above eq. becomes

; taking m and – 8 common

(m – 15)(m – 8)

**Question 8.**

Factorise.

m^{2} – 25m + 100

**Answer:**

On comparing with standard quadratic equation that is

we have,

a = 1, b = – 25 and c = 100

Now here,

Product a × c = 1 × 100 = 100

Factors of 100; 1 × 100; 2 × 50; 4 × 25;20 × 5

Sum should be b = – 25

From above factors ( – 20m – 5m)

will give – 25m sum

therefore – 25m is replaced by ( – 20m – 5m)

Now above eq. becomes

; taking m and – 5 common

(m – 5)(m – 20)

**Question 9.**

Factorise.

3x^{2} + 14x + 15

**Answer:**

On comparing with standard quadratic equation that is

we have,

a = 3, b = + 14 and c = + 15

Now here,

Product a × c = 3 × 15 = + 45

Factors of 45; 1 × 45; 5 × 9;15 × 3

Sum should be b = + 14

From above factors ( + 9x + 5x)

will give + 14x sum

therefore + 14x is replaced by ( + 9x + 5x)

Now above eq. becomes

; taking x and + 5 common

(x + 9)(x + 3)

**Question 10.**

Factorise.

2x^{2} + x – 45

**Answer:**

On comparing with standard quadratic equation that is

we have,

a = 2, b = 1 and c = – 45

Now here,

Product a × c = 2 × – 45 = 90

Factors of 90; 1 × 90; 2 × 45; 10 × 9; 30 × 3

Sum should be b = 1

From above factors ( + 10x – 9x)

will give + x sum

therefore + x is replaced by ( + 10x – 9x)

Now above eq. becomes

; taking 2x and – 9 common

(x + 5)(2x – 9)

**Question 11.**

Factorise.

20x^{2} – 26x + 8

**Answer:**

On comparing with standard quadratic equation that is

we have,

a = 20, b = – 26 and c = 8

Now here,

Product a × c = 20 × 8 = 160

Factors of 160; 2 × 80; 4 × 40; 8 × 20; 16 × 10; 32 × 5

Sum should be b = – 26x

From above factors ( – 16x – 10x)

will give – 26x sum

therefore – 26x is replaced by ( – 16x – 10x)

Now above eq. becomes

; taking 4x and – 2 common

2(2x – 1)(5x – 4)

**Question 12.**

Factorise.

44x^{2} – x – 3

**Answer:**

On comparing with standard quadratic equation that is

we have,

a = 44, b = – 1 and c = – 3

Now here,

Product a × c = – 132 = 44 × – 3

Factors of 132; 1 × 132; 2 × 66; 4 × 33;12 × 11

Sum should be b = – 1

From above factors ( – 12x – 11x)

will give – 1x sum

therefore – 1x is replaced by ( – 12x – 11x)

Now above eq. becomes

; taking x and – 9 common

(11x – 3)(4x – 1)

###### Practice Set 6.2

**Question 1.**

Factorise.

x^{3} + 64y^{3}

**Answer:**

We know that

- - - - - (i)

Here a = 1x, b = 4y; putting values in eq.i

⇒

⇒

}

Note: Must memorize cubes upto 12

**Question 2.**

Factorise.

125p^{3} + q^{3}

**Answer:**

We know that

- - - - - (i)

Here a = 5p, b = q; putting values in eq.i

⇒

⇒

}

Note: Must memorize cubes upto 12

**Question 3.**

Factorise.

125k^{3} + 27m^{3}

**Answer:**

We know that

- - - - - (i)

Here a = 5k, b = 3m; putting values in eq.i

⇒

⇒

⇒ }

}

}

Note: Must memorize cubes upto 12

**Question 4.**

Factorise.

2l^{3} + 432m^{3}

**Answer:**

We know that

- - - - - (i)

Taking 2 common, we get

Here a = l, b = 6m; putting values in eq.i

⇒

⇒

⇒ }]

}

Applying

}

Note: Must memorize cubes upto 12

**Question 5.**

Factorise.

24a^{3} + 81b^{3}

**Answer:**

We know that

- - - - - (i)

Taking 3 as common, we get

3; solving only bracket term first,

Here a = 2a, b = 3b; putting values in eq.i

}

Applying

}

}

Ans: - 3}

Note: Must memorize cubes upto 12

**Question 6.**

**Answer:**

We know that

- - - - - (i)

Here a = y, b = ; putting values in eq.i

Applying

}

Note: Must memorize cubes upto 12

**Question 7.**

Factorise.

**Answer:**

We know that

- - - - - (i)

Here a = a, b = ; putting values in eq.i

Applying

}

}

Note: Must memorize cubes upto 12

**Question 8.**

Factorise.

**Answer:**

We know that

- - - - - (i)

Here a = 1, b = ; putting values in eq.i

Applying

}

Note: Must memorize cubes upto 12

###### Practice Set 6.3

**Question 1.**

Factorise :

y^{3} – 27

**Answer:**

We know that

on comparison with above, we get

a = y, b = 3

Note: Must memorize cubes upto 12

**Question 2.**

Factorise :

x^{3} – 64y^{3}

**Answer:**

We know that

on comparison with above, we get

a = y, b = 3

Note: Must memorize cubes upto 12

**Question 3.**

Factorise :

27m^{3} – 216n^{3}

**Answer:**

We know that

on comparison with above, we get

a = 3m, b = 6n

Note: Must memorize cubes upto 12

**Question 4.**

Factorise :

125y^{3} – 1

**Answer:**

We know that

on comparison with above, we get

a = 5y, b = 1

Note: Must memorize cubes upto 12

**Question 5.**

Factorise :

8p^{3} – 27/p^{3}

**Answer:**

We know that

on comparison with above, we get

a = 2p, b = 3/p

Note: Must memorize cubes upto 12

**Question 6.**

Factorise :

343a^{3} – 512b^{3}

**Answer:**

We know that

on comparison with above, we get

a = 7a, b = 8b

Note: Must memorize cubes upto 12

**Question 7.**

Factorise :

64x^{2} – 729y^{2}

**Answer:**

We know that

on comparison with above, we get

a = 4x, b = 9y

Note: Must memorize cubes upto 12

**Question 8.**

Factorise :

16 a^{3} – 128/b^{3}

**Answer:**

We know that

taking 2 common from above given equation;

on comparison with above, we get

a = 2a, b = 4/b

Note: Must memorize cubes upto 12

**Question 9.**

Simplify :

(x + y)^{3} – (x – y)^{3}

**Answer:**

We know that

On comparing with given equation we get,

a = (3a + 5b), b = (3a – 5b)

Applying and

**Question 10.**

Simplify :

(3a + 5b)^{3} – (3a – 5b)^{3}

**Answer:**

We know that

On comparing with given equation we get,

a = (3a + 5b), b = (3a – 5b)

Applying and

**Question 11.**

Simplify :

(a + b)^{3} – a^{3} – b^{3}

**Answer:**

We know that

On comparing with given equation we get

**Question 12.**

Simplify :

p^{3} – (p + 1)^{3}

**Answer:**

We know that

On comparing with given equation we get

a = p, b = 1

**Question 13.**

Simplify :

(3xy – 2ab)^{3} – (3xy + 2ab)^{3}

**Answer:**

We know that

On comparing with given equation we get,

a = (3xy – 2ab), b = (3xy + 2ab)

Applying and

###### Practice Set 6.4

**Question 1.**

Simplify:

**Answer:**

We know that

applying these equation in above expression, we get

= 1

Note: - Try to factorize that term which help in reducing expression.

**Question 2.**

Simplify:

**Answer:**

We know that

and factorization of numerator and denominator

= a + 1

Note: - Try to factorize that term which help in reducing expression.

**Question 3.**

Simplify:

**Answer:**

We know that

and

Note: - Try to factorize that term which help in reducing expression.

**Question 4.**

Simplify:

**Answer:**

Applying and factorization, we get

= 1

Note: - Try to factorize that term which help in reducing expression.

**Question 5.**

Simplify:

**Answer:**

Applying

and factorization, we get, also changing by reversing N and D

Note: - Try to factorize that term which help in reducing expression.

**Question 6.**

Simplify:

**Answer:**

Applying

and factorization, we get

= x – 2

Note: - Try to factorize that term which help in reducing expression.

**Question 7.**

Simplify:

**Answer:**

Applying

_{,} factorization and we get, also changing by reversing N and D

= 5a + 1

Note: - Try to factorize that term which help in reducing expression.

**Question 8.**

Simplify:

**Answer:**

Applying

and factorization, we get

Note: - Try to factorize that term which help in reducing expression.