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

May 24, 2016

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

#### Explanation:

The simplest polynomial with these zeros is:

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

$= \left({x}^{2} - 9\right) \left({x}^{2} - 3 x + 2\right)$

$= {x}^{4} - 3 {x}^{3} - 7 {x}^{2} + 27 x - 18$

Standard form has one term of each degree in descending order of degree (omitting any terms with $0$ coefficient).

Any polynomial in $x$ with these zeros will be a multiple (scalar or polynomial) of this $f \left(x\right)$.