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

Dec 19, 2015

Begin with the factored form to find that one such polynomial is

$P \left(x\right) = {x}^{3} + 2 {x}^{2} - 13 x + 10$

#### Explanation:

A simple way of generating a polynomial with given roots is to begin with it in factored form. If $\left(x - {x}_{0}\right)$ is a factor of a polynomial $P \left(x\right)$, then $P \left({x}_{0}\right) = 0$ and so ${x}_{0}$ is a root of $P \left(x\right)$.

Applying this here, we get

$P \left(x\right) = \left(x - \left(- 5\right)\right) \left(x - 1\right) \left(x - 2\right)$

Again, plugging in any of the desired roots immediately yields $0$, meaning the polynomial has the desired properties. Then, all that remains is to multiply it out.

$P \left(x\right) = \left(x + 5\right) \left(x - 1\right) \left(x - 2\right)$

$= {x}^{3} + 2 {x}^{2} - 13 x + 10$