# How do you write a polynomial function given the real zeroes -2, -1, 0, 1, and 2 and coefficient 1?

Jan 2, 2016

$f \left(x\right) = {x}^{5} - 5 {x}^{3} + 4 x$

#### Explanation:

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

$= \left({x}^{2} - 4\right) \left({x}^{2} - 1\right) x$

$= \left({x}^{4} - 5 {x}^{2} + 4\right) x$

$= {x}^{5} - 5 {x}^{3} + 4 x$

This is the simplest polynomial in $x$ with the required zeros.

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

The polynomial graphed:

graph{x^5-5x^3+4x [-10, 10, -5, 5]}