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# factor theorem

If $f(x)$ is a polynomial over a ring with identity, then $x-a$ is a factor if and only if $a$ is a root (that is, $f(a)=0$).

This theorem is of great help for finding factorizations of higher degree polynomials. As example, let us think that we need to factor the polynomial $p(x)=x^{3}+3x^{2}-33x-35$. With some help of the rational root theorem we can find that $x=-1$ is a root (that is, $p(-1)=0$), so we know $(x+1)$ must be a factor of the polynomial. We can write then

$p(x)=(x+1)q(x)$ |

where the polynomial $q(x)$ can be found using long or synthetic division of $p(x)$ between $x-1$. In our case $q(x)=x^{2}+2x-35$ which can be easily factored as $(x-5)(x+7)$. We conclude that

$p(x)=(x+1)(x-5)(x+7).$ |

Related:

Polynomial,RationalRootTheorem,Root, APolynomialOfDegreeNOverAFieldHasAtMostNRoots

Synonym:

root theorem

Type of Math Object:

Theorem

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

12D10*no label found*12D05

*no label found*

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