Q. For what value of k are the roots of the quadratic polynomial kx(x – 2) + 6 = 0 are equal?

  • Sol. Given,
  • kx(x – 2) + 6 = 0
  • ⇒ kx2 – 2kx + 6 = 0
  • Let α and β be the roots of the polynomial.
302

Q. Find a cubic polynomial when the sum, sum of the products of its zeroes taken two at a time and product of its zero are 2, – 7, – 14 respectively.

  • Ans. Let the zeroes of the cubic polynomial be α, β and γ.
  • α + β + γ = 2
  • αβ+ βγ + αγ = – 7
  • αβ γ = – 14
  • We know, a cubic polynomial is given as,
  • f (x) = x3 – (α + β + γ)x2
  • + (αβ + βγ + αγ)x – αβ γ
  • f (x) = x3 – 2x2 – 7x + 14
  • Thus, the required cubic polynomial is
  • x3 – 2x2 – 7x + 14. Ans
Q. Verify that 3, – 1 and  -1/3 are the zeroes of the cubic polynomial p(x) = 3x3 – 5x2 – 11x – 3 and also verify the relationship between the zeroes and their coefficients.
  • Sol. In order to verify that 3, – 1 and −1/3
  • zeroes of the cubic polynomial
  • p(x) = 3x3 – 5x2 – 11x – 3,
  • we have to substitute x = 3, – 1, -1/3 in p(x) and ensure that the result is 0.
  • Thus p(x) = 3x3 – 5x2 – 11x – 3
  • p(3) = 3(3)3 – 5(3)2 – 11 × 3 – 3
  • = 3 × 27 – 5 × 9 – 33 – 3
  • = 81 – 45 – 36
  • = 81 – 81 = 0
  • p(– 1) = 3(– 1)3 – 5(– 1)2 – 11(– 1) – 3
  • = – 3 – 5 + 11 – 3
  • = – 11 + 11 = 0
304

Hence, the relationship between the zeroes and the their coefficients has also been verified.
Hence Verified.