# proof of Vieta’s formula

Proof: We can write $P(x)$ as

$$P(x)={a}_{n}(x-{r}_{1})(x-{r}_{2})\mathrm{\dots}(x-{r}_{n})={a}_{n}({x}^{n}-{S}_{1}{x}^{n-1}+{S}_{2}{x}^{n-2}-\mathrm{\dots}+{(-1)}^{n}{S}_{n}).$$ |

Since we also have that

$$P(x)={a}_{n}{x}^{n}+{a}_{n-1}{x}^{n-1}+\mathrm{\dots}{a}_{1}x+{a}_{0},$$ |

setting the coefficients equal yields

$${a}_{n}{(-1)}^{i}{S}_{i}={a}_{n-i}$$ |

which is what the theorem stated.

Title | proof of Vieta’s formula |
---|---|

Canonical name | ProofOfVietasFormula |

Date of creation | 2013-03-22 15:26:59 |

Last modified on | 2013-03-22 15:26:59 |

Owner | neapol1s (9480) |

Last modified by | neapol1s (9480) |

Numerical id | 4 |

Author | neapol1s (9480) |

Entry type | Proof |

Classification | msc 12Y05 |