Question

Find a cubic polynomial with the sum, sum of the products of its zeros taken two at a time, and product of its zeros as `2, -7, -14` respectively.

Answer 1

Let `alpha, beta, gamma ` be the zeros of the required polynomial. Then,
` alpha+beta+gamma = 2, alpha beta+beta gamma+gamma alpha =- 7 and alpha beta gamma =- 14.`
So, the required polynomial is
`p(x) = x^(3) -(alpha+beta+gamma) x^(2) + (alpha beta+beta gamma+gamma alpha) x - alpha beta gamma`
` = x^(3) - 2x^(2) - 7x-(-14) = x^(3) - 2x^(2) - 7x+14.`