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factor theorem (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

\begin{displaymath}p(x)=(x+1)q(x)\end{displaymath}

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
\begin{displaymath}p(x)=(x+1)(x-5)(x+7).\end{displaymath}



"factor theorem" is owned by drini. [ full author list (3) | owner history (1) ]
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See Also: polynomial, rational root theorem, root, a polynomial of degree $n$ over a field has at most $n$ roots

Other names:  root theorem

Attachments:
proof of factor theorem (Proof) by Wkbj79
proof of factor theorem using division (Proof) by alozano
proof of factor theorem due to Fermat (Proof) by pahio
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Cross-references: synthetic division, rational root theorem, degree, root, factor, identity, ring, polynomial
There are 6 references to this entry.

This is version 7 of factor theorem, born on 2002-02-04, modified 2008-08-28.
Object id is 1810, canonical name is FactorTheorem.
Accessed 9821 times total.

Classification:
AMS MSC: 12D05 (Field theory and polynomials :: Real and complex fields :: Polynomials: factorization)
 12D10 (Field theory and polynomials :: Real and complex fields :: Polynomials: location of zeros )
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