Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Vieta's Formulas

Revision as of 23:26, 18 June 2006 by Me@home (talk | contribs)

Background

Let $P(x)={a_n}x^n+{a_{n-1}}x^{n-1}+\cdots+{a_1}x+a_0$, where the coefficient of $x^{i}$ is ${a}_i$. As a consequence of the Fundamental Theorem of Algebra, we can also write $P(x)=a_n(x-r_1)(x-r_2)\cdots(x-r_n)$, where ${r}_i$ are the roots of $P(x)$.

Also, let ${\sigma}_k$ be the ${}{k}$th symmetric sum.

Statement

$\sigma_k = (-1)^k\cdot \frac{a_{n-k}}{a_n{}}$, for ${}1\le k\le {n}$.

Proof

[needs to be added]

Related Links

Mathworld's Article

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Vieta%27s_Formulas&oldid=2975"