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单词 FactorTheorem
释义

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

p(x)=(x+1)q(x)

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

p(x)=(x+1)(x-5)(x+7).
Titlefactor theorem
Canonical nameFactorTheorem
Date of creation2013-03-22 12:17:24
Last modified on2013-03-22 12:17:24
Ownerdrini (3)
Last modified bydrini (3)
Numerical id10
Authordrini (3)
Entry typeTheorem
Classificationmsc 12D10
Classificationmsc 12D05
Synonymroot theorem
Related topicPolynomial
Related topicRationalRootTheorem
Related topicRoot
Related topicAPolynomialOfDegreeNOverAFieldHasAtMostNRoots
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更新时间:2025/5/4 21:29:22