factor theoremFactorTheorem2002-02-04 01:43:492008-08-28 06:40:27Theorem%\usepackage{graphicx}
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\newcommand{\mathbb}[1]{\mathbbmss{#1}}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
$$p(x)=(x+1)(x-5)(x+7).$$