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