# How do you find the polynomial function with roots 2, 3+- sqrt2?

Dec 5, 2015

Start from the factored form to find the desired polynomial to be
$f \left(x\right) = {x}^{3} - 8 {x}^{2} + 19 x - 14$

#### Explanation:

Creating a polynomial function with specific roots is as easy as can be. Just set it up as a product of binomials $\left(x - \text{root}\right)$ which evaluate to $0$ at the desired root. For example, in this case, we would use

$f \left(x\right) = \left(x - 2\right) \left(x - \left(3 + \sqrt{2}\right)\right) \left(x - \left(3 - \sqrt{2}\right)\right)$

$= {x}^{3} - 8 {x}^{2} + 19 x - 14$

Because we start from the fully factored form, it is clear that the roots are as desired.