Question

How do you find the polynomial function with roots `2 + 3sqrt3`?

Answer 1

`f(x)=x^2-4x-23`

Explanation:

If a function has the root `a+sqrtb`, it must also have the root `a-sqrtb`.

So, this function actually has the roots `2+3sqrt3` and `2-3sqrt3`. Since these are roots, when `x` takes on these values, `f(x)=0`. Thus, each factor of the polynomial will be `x -`the root.

`f(x)=(x-(2+3sqrt3))(x-(2-3sqrt3))`

`f(x)=(x-2-3sqrt3)(x-2+3sqrt3)`

`f(x)=(x-2)^2+(-3sqrt3)(3sqrt3)`

`f(x)=x^2-4x+4-27`

`f(x)=x^2-4x-23`