factor theorem

If f(x) is a polynomialMathworldPlanetmathPlanetmathPlanetmath 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)=x3+3x2-33x-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


where the polynomial q(x) can be found using long or synthetic divisionMathworldPlanetmath of p(x) between x-1. In our case q(x)=x2+2x-35 which can be easily factored as (x-5)(x+7). We conclude that

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