Question

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

`sqrt2 , 1/3`

Answer 1

Given: α + β = `sqrt2`, αβ = `1/3`

Since ax² + bx + c = kx² - k(α + β)x + kαβ

or ax² + bx + c = k[x² - (α + β)x + αβ]

Or `(ax^2 + bx + c)/k = (x^2 - sqrt2x + 1/3)`

Or `(ax^2 + bx + c)/k = (3x^2 - 3sqrt2x + 1)/3`

Here k is a constant term, by comparing k = 3

Hence, ax² + bx + c = `3x^2 - 3sqrt2x + 1`

The quadratic polynomial is `3x^2 - 3sqrt2x + 1`.