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)=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
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)=x2+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 |