proof of factor theorem due to Fermat

Lemma (cf. factor theorem).  If the polynomialMathworldPlanetmathPlanetmathPlanetmath


vanishes at  x=c,  then it is divisible by the differencePlanetmathPlanetmath x-c, i.e. there is valid the identic equation

f(x)(x-c)q(x) (1)

where q(x) is a polynomial of degree n-1, beginning with the a0xn-1.

The lemma is here proved by using only the properties of the multiplicationPlanetmathPlanetmath and addition, not the division.

Proof.  If we denote  x-c=y,  we may write the given polynomial in the form


It’s clear that every (y+c)k is a polynomial of degree k with respect to y, where yk has the coefficient 1 and the is ck.  This implies that f(x) may be written as a polynomial of degree n with respect to y, where yn has the coefficient a0 and the on y is equal to  a0cn+a1cn-1++an-1c+an, i.e. f(c).  So we have


where  b1,b2,,bn-1  are certain coefficients.  If we return to the indeterminateMathworldPlanetmath x by substituting in the last identic equation x-c for y, we get


When the powers (x-c)k are expanded to polynomials, we see that the expression in the brackets is a polynomial q(x) of degree  n-1  with respect to x and with the coefficient a0 of xn-1.  Thus we obtain

f(x)f(c)+(x-c)q(x). (2)

This result is true independently on the value of c.  If this value is chosen such that  f(c)=0,  then (2) reduces to (1), Q. E. D.


  • 1 Ernst Lindelöf: Johdatus korkeampaan analyysiin (‘Introduction to Higher Analysis’).  Fourth edition. WSOY, Helsinki (1956).
Title proof of factor theorem due to Fermat
Canonical name ProofOfFactorTheoremDueToFermat
Date of creation 2013-03-22 15:40:12
Last modified on 2013-03-22 15:40:12
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 10
Author pahio (2872)
Entry type Proof
Classification msc 12D10
Classification msc 12D05
Synonym proof of factor theorem without division