Question

How do you find a polynomial function that has zeros `2, 4+sqrt5, 4-sqrt5`?

Answer 1

`f(x) = x^3-10x^2+27x-22`

Explanation:

If `x=a` is a zero then `(x-a)` is a factor.

So the simplest polynomial function of `x` that has these zeros is:

`f(x) = (x-2)(x-(4+sqrt(5)))(x-(4-sqrt(5)))`

`color(white)(f(x)) = (x-2)((x-4)+sqrt(5))((x-4)-sqrt(5))`

`color(white)(f(x)) = (x-2)((x-4)^2-5)`

`color(white)(f(x)) = (x-2)(x^2-8x+16-5)`

`color(white)(f(x)) = (x-2)(x^2-8x+11)`

`color(white)(f(x)) = x^3-10x^2+27x-22`

Any polynomial in `x` with these zeros will be a multiple (scalar or polynomial) of this `f(x)`.

`color(white)()`
Footnote

In this example, we were asked for a function with zeros including both `4+sqrt(5)` and its radical conjugate `4-sqrt(5)`. If only one had been specified then we would still have had to include the other if we wanted our function to have rational coefficients.