Back to the index. Or to the chambers

This article has 4 links. View as Cloud or List.

Root Basic Polynomial Order And Degree Of Polynomial Quadratic Formula

1	Root
1	Basic Polynomial
3	Order And Degree Of Polynomial
5	Quadratic Formula

No pages link here.

Planetmath Browser (2008—2009)

BSD licence | A django site

All articles taken from PlanetMath.org snapshot under CC-BY-SA licence.

→ The original article on PlanetMath.org

Other Formats: LaTeX

New! You can click on formulas to copy the LaTeX source to your clipboard. | Math Videos

Vietas Formula

Suppose is a polynomial of degree with roots $r_1, r_2, \ldots, r_n$ (not necessarily distinct). For $1\leq k\leq n$ , define by

$\displaystyle S_k = \sum\limits_{1\leq\alpha_{1} < \alpha_{2} < \ldots\alpha_k\leq n} r_{\alpha_1}r_{\alpha_2}\ldots r_{\alpha_k}$

For example,

$\displaystyle S_1 = r_1 + r_2 + r_3 + \ldots + r_n$

$\displaystyle S_2 = r_1r_2 + r_1r_3 + r_1r_4 + r_2r_3 + \ldots + r_{n-1}r_{n}$

Then writing

$\displaystyle P(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots a_{1}x + a_{0},$

we find that

$\displaystyle S_i = (-1)^{i}\frac{a_{n-i}}{a_n}$

For example, if is a polynomial of degree 1, then and clearly $r_1 = -\frac{a_0}{a_1}$ .

If is a polynomial of degree 2, then and $r_1 + r_2 = -\frac{a_1}{a_2}$ and $r_1r_2 = \frac{a_0}{a_2}$ . Notice that both of these formulas can be determined from the quadratic formula.

More intrestingly, if , then $r_1 + r_2 + r_3 = -\frac{a_2}{a_3}$ , $r_1r_2 + r_2r_3 + r_3r_1 = \frac{a_1}{a_3}$ , and $r_1r_2r_3 = -\frac{a_0}{a_3}$ .