# proof of factor theorem due to Fermat

Lemma (cf. factor theorem). If the polynomial^{}

$$f(x):={a}_{0}{x}^{n}+{a}_{1}{x}^{n-1}+\mathrm{\cdots}+{a}_{n-1}x+{a}_{n}$$ |

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

$f(x)\equiv (x-c)q(x)$ | (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

$$f(x)={a}_{0}{(y+c)}^{n}+{a}_{1}{(y+c)}^{n-1}+\mathrm{\cdots}+{a}_{n-1}(y+c)+{a}_{n}.$$ |

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 ${a}_{0}{c}^{n}+{a}_{1}{c}^{n-1}+\mathrm{\cdots}+{a}_{n-1}c+{a}_{n}$, i.e. $f(c)$. So we have

$$f(x)={a}_{0}{y}^{n}+{b}_{1}{y}^{n-1}+{b}_{2}{y}^{n-2}+\mathrm{\cdots}+{b}_{n-1}y+f(c)=f(c)+y\cdot ({a}_{0}{y}^{n-1}+{b}_{1}{y}^{n-2}+\mathrm{\cdots}+{b}_{n-1}+{a}_{n}),$$ |

where ${b}_{1},{b}_{2},\mathrm{\dots},{b}_{n-1}$ are certain coefficients. If we return to the indeterminate^{} $x$ by substituting in the last identic equation $x-c$ for $y$, we get

$$f(x)\equiv f(c)+(x-c)[{a}_{0}{(x-c)}^{n-1}+{b}_{1}{(x-c)}^{n-2}+\mathrm{\cdots}+{b}_{n-1}].$$ |

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

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

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 |