# Polynomials

**Exercise 1**

**1.** The graphs of y=p(x) are given to us, for some polynomials p(x). Find the number of zeroes of p(x), in each case.

**Exercise 2**

**2**. Find the zeroes of the following quadratic polynomials and verify the relationship between the zeros and the coefficients.

(i) x^{2}-2x-8

(ii) 4s^{2}-4s+1

(iii) 6x^{2}-3-7x

(iv) 4u^{2}+8u

(v) t^{2}-15

(vi) 3x^{2}-x-4

**3.** Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.

**Exercise 3**

**4.** Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following.

(i) p(x) = x^{3}-3x^{2}+5x-3, g(x) = x^{2}-2

(ii) p(x) = x^{4}-3x^{2}+4x+5, g(x) = x^{2}-x+1

(iii) p(x) = x^{4}-5x+6, g(x) = 2-x^{2}

**5.** Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial.

(i) t^{2}-3, 2t^{4}+3t^{3}-2t^{2}-9t-12

(ii) x^{2}+3x+1, 3x^{4}+5x^{3}-7x^{2}+2x+2

(iii) x^{3}-3x+1, x^{5}-4x^{3}+x^{2}+3x+1

**6**. Obtain all other zeroes of (3x^{4}+6x^{3}-2x^{2}-10x-5), if two of its zeroes are

**7**. On dividing (x^{3}-3x^{2}+x+2) by a polynomial g(x), the quotient and remainder were (x-2) and (-2x+4) respectively. Find g(x).

**8.**Give examples of polynomials p(x), g(x), q(x) and r(x), which satisfy the division algorithm and

(i) deg p(x) = deg q(x)

(ii) deg q(x) = deg r(x)

(iii) deg r(x) = 0

**Exercise 4**

**9.** Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes and the coefficients in each case:

**10.** Find a cubic polynomial with the sum of the product of its zeroes taken two at a time and the product of its zeroes are 2, -7, -14 respectively.

**11.** If the zeroes of the polynomial *x ^{3}* -3x

^{2}+ x +1 are

*a-b, a, a+b,*find

*a*and

*b.*

**12.** If the two zeroes of the polynomial x^{4} – 6x^{3} – 26x^{2} +138x -35 are 2 ± √3, find other zeroes.

**13.** If the polynomial x^{4} – 6x^{3} +16x^{2} – 25x +10 is divided by another polynomial x^{2} – 2x + *k, *the remainder comes out to be x *+a, *find *k *and *a.*