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

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

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Answer 1 · 2015-09-24T19:12:14

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 $a \cdot b = 0$ $a*b = 0$ , then either $a = 0$ $a = 0$ or $b = 0$ $b = 0$ or both.
Example: Find the roots of the polynomial by factoring:
$P (x) = x^{3} - x^{2} - x + 1$ $P(x) = x^3-x^2-x+1$ , set to zero:
$x^{3} - x^{2} - x + 1 = 0$ $x^3-x^2-x+1=0$ , factor by grouping:
$x^{2} (x - 1) - 1 (x - 1) = 0$ $x^2(x-1)-1(x-1)=0$
$(x^{2} - 1) (x - 1) = 0$ $(x^2-1)(x-1)=0$ , use difference of squares to factor further:
$(x + 1) (x - 1) (x - 1) = 0$ $(x+1)(x-1)(x-1)=0$ , use the zero factor property:
$x + 1 = 0 \Rightarrow x = - 1$ $x+1=0=>x=-1$
$x - 1 = 0 \Rightarrow x = 1$ $x-1=0=>x=1$
Notice that $x = 1$ $x = 1$ has a multiplicity of 2.

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

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