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

 $p(x)=(x+1)(x-5)(x+7).$
 Title factor theorem Canonical name FactorTheorem Date of creation 2013-03-22 12:17:24 Last modified on 2013-03-22 12:17:24 Owner drini (3) Last modified by drini (3) Numerical id 10 Author drini (3) Entry type Theorem Classification msc 12D10 Classification msc 12D05 Synonym root theorem Related topic Polynomial Related topic RationalRootTheorem Related topic Root Related topic APolynomialOfDegreeNOverAFieldHasAtMostNRoots