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

Jul 13, 2015

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

#### Explanation:

If $k$ is a root of a polynomial function $f \left(x\right)$
then $f \left(k\right) = 0$

Obviously, if $\left(x - a\right)$ is a factor of $f \left(x\right)$ then $f \left(a\right) = 0$ and thus $a$ is a root of $f \left(x\right)$.

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.