# proof of factor theorem due to Fermat

Lemma (cf. factor theorem).  If the polynomial

 $f(x):=a_{0}x^{n}\!+\!a_{1}x^{n-1}\!+\cdots+\!a_{n-1}x\!+\!a_{n}$

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

 $\displaystyle f(x)\equiv(x\!-\!c)q(x)$ (1)

where $q(x)$ is a polynomial of degree $n\!-\!1$, beginning with the $a_{0}x^{n-1}$.

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

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

 $f(x)=a_{0}(y\!+\!c)^{n}\!+\!a_{1}(y\!+\!c)^{n-1}\!+\cdots+\!a_{n-1}(y\!+\!c)\!% +\!a_{n}.$

It’s clear that every $(y\!+\!c)^{k}$ is a polynomial of degree $k$ with respect to $y$, where $y^{k}$ has the coefficient 1 and the is $c^{k}$.  This implies that $f(x)$ may be written as a polynomial of degree $n$ with respect to $y$, where $y^{n}$ has the coefficient $a_{0}$ and the on $y$ is equal to  $a_{0}c^{n}\!+\!a_{1}c^{n-1}\!+\cdots+\!a_{n-1}c\!+\!a_{n}$, i.e. $f(c)$.  So we have

 $f(x)=a_{0}y^{n}\!+\!b_{1}y^{n-1}\!+\!b_{2}y^{n-2}\!+\cdots+\!b_{n-1}y\!+f(c)=f% (c)+y\cdot(a_{0}y^{n-1}\!+\!b_{1}y^{n-2}\!+\cdots+\!b_{n-1}\!+\!a_{n}),$

where  $b_{1},\,b_{2},\,\ldots,\,b_{n-1}$  are certain coefficients.  If we return to the indeterminate $x$ by substituting in the last identic equation $x\!-\!c$ for $y$, we get

 $f(x)\equiv f(c)+(x\!-\!c)[a_{0}(x\!-\!c)^{n-1}\!+\!b_{1}(x\!-\!c)^{n-2}\!+% \cdots+\!b_{n-1}].$

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 $a_{0}$ of $x^{n-1}$.  Thus we obtain

 $\displaystyle f(x)\equiv f(c)+(x\!-\!c)q(x).$ (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.

## References

• 1 Ernst Lindelöf: Johdatus korkeampaan analyysiin (‘Introduction to Higher Analysis’).  Fourth edition. WSOY, Helsinki (1956).
Title proof of factor theorem due to Fermat ProofOfFactorTheoremDueToFermat 2013-03-22 15:40:12 2013-03-22 15:40:12 pahio (2872) pahio (2872) 10 pahio (2872) Proof msc 12D10 msc 12D05 proof of factor theorem without division