Question

How do you find the polynomial function with roots 1, -2, and 5?

Answer 1

A polynomial function with roots 1, -2, and 5 would be
`f(x)=(x-1)(x+2)(x-5)`
Note however that there are infinitely many polynomial function with these roots.

Explanation:

If `k` is a root of a polynomial function `f(x)`
then `f(k) = 0`

Obviously, if `(x-a)` is a factor of `f(x)` then `f(a) = 0` and thus `a` is a root of `f(x)`.

Although there are infinitely many polynomials with the specified roots, all such polynomials in x will be multiples of f(x), i.e. they will have f(x) as a factor.