# How does the zero factor property relate to factoring a polynomial?

Sep 24, 2015

Well, if you a polynomial is factorable then its roots/zeroes can be easily found by setting it to zero and using the zero factor property. Please see explanation below.

#### Explanation:

The Zero Product Property:
A product of factors is zero if and only if one or more of the factors is zero. Or:
if $a \cdot b = 0$, then either $a = 0$ or $b = 0$ or both.
Example: Find the roots of the polynomial by factoring:
$P \left(x\right) = {x}^{3} - {x}^{2} - x + 1$, set to zero:
${x}^{3} - {x}^{2} - x + 1 = 0$, factor by grouping:
${x}^{2} \left(x - 1\right) - 1 \left(x - 1\right) = 0$
$\left({x}^{2} - 1\right) \left(x - 1\right) = 0$, use difference of squares to factor further:
$\left(x + 1\right) \left(x - 1\right) \left(x - 1\right) = 0$, use the zero factor property:
$x + 1 = 0 \implies x = - 1$
$x - 1 = 0 \implies x = 1$
Notice that $x = 1$ has a multiplicity of 2.