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Homeproof of factor theorem due to Fermat

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# 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 term $a_{0}x^{{n-1}}$.

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 constant term 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 term independent 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).

## Mathematics Subject Classification

12D10*no label found*12D05

*no label found*

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