Question

Find the quadratic polynomial, the sum of whose zeros is `sqrt2` and their product is `-12`. Hence, find the zeros of the polynomial.

Answer 1

Let `alpha and beta` be the zeros of the required polynomial f(x).
Than, `(alpha+beta) = sqrt2 and alpha beta =- 12.`
`:. F(x) = x^(2) - (alpha+beta) x + alpha beta `
` = x^(2) - sqrt 2 x - 12.`
So, the required polynomial is `f(x) = x(2) - sqrt2 x - 12.`
Now, `f(x) = x^(2) - sqrt2 x - 12`
`= x^(2) - 3sqrt2 x + 2sqrt2 x - 12` [note it]
` = x(x-3sqrt2)+2sqrt2(x-3sqrt2)`
` = (x-3sqrt2)(x+2sqrt2).`
` :. f(x) = 0 rArr (x-3sqrt2)(x+2sqrt2) = 0`
` rArr x -3sqrt2 = 0 or x + 2 sqrt2 = 0`
` rArr x = 3 sqrt2 or x =- 2sqrt2.`
Hence, the required polynomial is ` f(x) = x^(2) - sqrt2x - 12` whose zeros are `3sqrt2 and -2sqrt2.`