Question

If a polynomial function with rational coefficients has the zeros 2, `-2+sqrt10`, what are the additional zeros?

Answer 1

`-2-sqrt(10)` must also be a zero and the function will be a multiple (scalar or polynomial) of:

`f(x) = x^3+2x^2-14x+12`

Explanation:

Given that `2` and `-2+sqrt(10)` are zeros, the rational conjugate `-2-sqrt(10)` must also be a zero.

The polynomial function will be a multiple (scalar or polynomial of this `f(x)`:

`f(x) = (x-2)(x+2-sqrt(10))(x+2+sqrt(10))`

`color(white)(f(x)) = (x-2)((x+2)-sqrt(10))((x+2)+sqrt(10))`

`color(white)(f(x)) = (x-2)((x+2)^2-(sqrt(10))^2)`

`color(white)(f(x)) = (x-2)(x^2+4x+4-10)`

`color(white)(f(x)) = (x-2)(x^2+4x-6)`

`color(white)(f(x)) = x^3+2x^2-14x+12`