proof of factor theorem using division

Lemma (cf. factor theorem).

Let R be a commutative ring with identityPlanetmathPlanetmath and let p(x)R[x] be a polynomialMathworldPlanetmathPlanetmathPlanetmath with coefficients in R. The element aR is a root of p(x) if and only if (x-a) divides p(x).


Let p(x) be a polynomial in R[x] and let a be an element of R.

  1. 1.

    First we assume that (x-a) divides p(x). Therefore, there is a polynomial q(x)R[x] such that p(x)=(x-a)q(x). Hence, p(a)=(a-a)q(a)=0 and a is a root of p(x).

  2. 2.

    Assume that a is a root of p(x), i.e. p(a)=0. Since x-a is a monic polynomialMathworldPlanetmath, we can perform the polynomial long division ( of p(x) by (x-a). Thus, there exist polynomials q(x) and r(x) such that:


    and the degree of r(x) is less than the degree of x-a (so r(x) is just a constant). Moreover, 0=p(a)=0+r(a)=r(a)=r(x). Therefore p(x)=(x-a)q(x) and (x-a) divides p(x).

Title proof of factor theorem using division
Canonical name ProofOfFactorTheoremUsingDivision
Date of creation 2013-03-22 15:08:58
Last modified on 2013-03-22 15:08:58
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 8
Author alozano (2414)
Entry type Proof
Classification msc 12D10
Classification msc 12D05