# How do you write a polynomial in standard form given the zeros 3, 1, 2, and-3?

Jun 29, 2018

$f \left(x\right) = {x}^{4} - 3 {x}^{3} - 7 {x}^{2} + 27 x - 18$

#### Explanation:

Using $x$ as our variable name, each zero $a$ corresponds to a factor $\left(x - a\right)$.

So we can write:

$f \left(x\right) = \left(x - 3\right) \left(x - 1\right) \left(x - 2\right) \left(x + 3\right)$

$\textcolor{w h i t e}{f \left(x\right)} = \left(\left(x - 3\right) \left(x + 3\right)\right) \left(\left(x - 1\right) \left(x - 2\right)\right)$

$\textcolor{w h i t e}{f \left(x\right)} = \left({x}^{2} - 9\right) \left({x}^{2} - 3 x + 2\right)$

$\textcolor{w h i t e}{f \left(x\right)} = {x}^{2} \left({x}^{2} - 3 x + 2\right) - 9 \left({x}^{2} - 3 x + 2\right)$

$\textcolor{w h i t e}{f \left(x\right)} = \left({x}^{4} - 3 {x}^{3} + 2 {x}^{2}\right) - \left(9 {x}^{2} - 27 x + 18\right)$

$\textcolor{w h i t e}{f \left(x\right)} = {x}^{4} - 3 {x}^{3} - 7 {x}^{2} + 27 x - 18$

This is the simplest polynomial with the given zeros. Any polynomial in $x$ with these zeros is a multiple (scalar or polynomial) of this $f \left(x\right)$.