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.
2. Find the zeroes of the following quadratic polynomials and verify the relationship between the zeros and the coefficients.
3. Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
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) = x3-3x2+5x-3, g(x) = x2-2
(ii) p(x) = x4-3x2+4x+5, g(x) = x2-x+1
(iii) p(x) = x4-5x+6, g(x) = 2-x2
5. Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial.
(i) t2-3, 2t4+3t3-2t2-9t-12
(ii) x2+3x+1, 3x4+5x3-7x2+2x+2
(iii) x3-3x+1, x5-4x3+x2+3x+1
7. 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).
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
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 x3 -3x2 + x +1 are a-b, a, a+b, find a and b.
12. If the two zeroes of the polynomial x4 – 6x3 – 26x2 +138x -35 are 2 ± √3, find other zeroes.
13. If the polynomial x4 – 6x3 +16x2 – 25x +10 is divided by another polynomial x2 – 2x + k, the remainder comes out to be x +a, find k and a.