### Polynomials Class 10th Mathematics AP Board Solution

##### Question 1.Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following:p(x) = x3 – 3x2 + 5x – 3, g(x) = x2 - 2Answer:p(x) = x3 – 3x2 + 5x – 3g(x) = x2 – 2 On dividing them,The quotient is x – 3.And,The remainder is 7x – 9.Question 2.Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following:p(x) = x4 – 3x2 + 4x + 5, g(x) = x2 + 1 - xAnswer:p(x) = x4 – 3x2 + 4x + 5g(x) = x2 + 1 – xOn dividing them, The quotient is x2 + x - 3And,The remainder is 8Question 3.Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following:p(x) = x4 – 5x + 6, g(x) = 2 - x2Answer:p(x) = x4 – 5x + 6g(x) = 2 – x2On dividing them, The quotient is –x2 – 2And,The remainder is -5x + 10Question 4.Check in which case the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial.t2 – 3, 2t4 + 3t3 – 2t2 – 9t - 12Answer:p(x) = 2t4 + 3t3 – 2t2 - 9t – 12g(x) = t2 – 3On dividing them, The remainder is 0Hence, yes t2 – 3 is a factor of 2t4 + 3t3 – 2t2 - 9t – 12Question 5.Check in which case the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial.x2 + 3x + 1, 3x4 + 5x3 – 7x2 + 2x + 2Answer:p(x) = 3x4 + 5x3 – 7x2 + 2x + 2g(x) = x2 + 3x + 1On dividing them, The remainder is 0Hence, yes x2 + 3x + 1 is a factor of 3x4 + 5x3 – 7x2 + 2x + 2Question 6.Check in which case the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial.x2 - 3x + 1, x5 - 4x3 + x2 + 3x + 1Answer:p(x) = x5 – 4x3 + x2 + 3x + 1g(x) = x3 – 3x + 1On dividing them, The remainder is 2 ≠ 0Hence, no x3 – 3x + 1 is not a factor of x5 – 4x3 + x2 + 3x + 1Question 7.Obtain all other zeroes of 3x4 + 6x3 – 2x2 – 10x – 5,if two of its zeroes are and Answer:Two zeroes are and  is a factor is a factor is a factorNow, Therefore, 3x2 + 6x +3 is also a factorDividing 3x2 + 6x +3 by 3,We get,x2 + 2x +1Factorising x2 + 2x +1,x2 + 2x +1 = 0⇒ x2 + x + x +1 = 0⇒ x(x+ 1) + 1(x +1) = 0⇒ (x+ 1)(x +1) = 0⇒ x= -1, -1Therefore, the other zeroes are -1 and -1.Question 8.On dividing x3 – 3x2 + x + 2 by a polynomial g(x), the quotient and remainder were x - 2 and – 2x + 4 respectively. Find g(x)Answer:Dividend = x3 – 3x2 + x + 2Quotient = x – 2Remainder = -2x + 4Dividend = Divisor × Quotient + Remainder⇒ x3 – 3x2 + x + 2 = Divisor ×(x – 2) + (-2x + 4)⇒ x3 – 3x2 + x + 2 – (-2x + 4) = Divisor ×(x – 2)⇒ x3 – 3x2 + x + 2 + 2x – 4 = Divisor ×(x – 2)⇒ x3 – 3x2 + 3x – 2 = Divisor ×(x – 2) Therefore, g(x) = x2 – x + 1Question 9.Give examples of polynomials p(x), g(x), q(x) and r(x) which satisfy the division algorithm anddeg p (x) = deg q (x)Answer:Let g(x) = 2And p(x) = 2x2 – 2x + 14Then, dividing p(x) by g(x) gives.q(x) = x2 – x + 7and,r(x) = 0Deg(p(x)) = Deg(q(x)) = 2Hence,g(x) = 2p(x) = 2x2 – 2x + 14q(x) = x2 – x + 7r(x) = 0Question 10.Give examples of polynomials p(x), g(x), q(x) and r(x) which satisfy the division algorithm anddeg q (x) = deg r (x)Answer:Let g(x) = x + 1And p(x) = x2 + 3Then, dividing p(x) by g(x) givesq(x) = xand,r(x) = -x + 3Deg(r(x)) = Deg(q(x)) = 1Hence,g(x) = x + 1p(x) = x2 + 3q(x) = xr(x) = -x + 3Question 11.Give examples of polynomials p(x), g(x), q(x) and r(x) which satisfy the division algorithm anddeg r (x) = 0Answer:Let g(x) = 3And p(x) = 3x + 3Then, dividing p(x) by g(x) givesq(x) = x + 1and,r(x) = 0Deg(r(x)) = 0Hence,g(x) = 3p(x) = 3x + 3q(x) = x + 1r(x) = 0

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