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Vieta's formula (Theorem)

Suppose $ P(x)$ is a polynomial of degree $ n$ with roots $ r_1, r_2, \ldots, r_n$ (not necessarily distinct). For $ 1\leq k\leq n$, define $ S_k$ 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 $ P(x)$ as
$\displaystyle P(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots a_{1}x + a_{0},$
we find that
$\displaystyle S_i = (-1)^{i}\frac{a_{n-i}}{a_n}$

For example, if $ P(x)$ is a polynomial of degree 1, then $ P(x) = a_1x + a_0$ and clearly $ r_1 = -\frac{a_0}{a_1}$.

If $ P(x)$ is a polynomial of degree 2, then $ P(x) = a_2x^2 + a_1x + a_0$ 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 $ P(x) = a_3x^3 + a_2x^2 + a_1x + a_0$, 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}$.



"Vieta's formula" is owned by neapol1s.
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See Also: properties of quadratic equation


Attachments:
proof of Vieta's formula (Proof) by neapol1s

Cross-references: quadratic formula, formulas, roots, degree, polynomial

This is version 6 of Vieta's formula, born on 2005-06-25, modified 2006-01-12.
Object id is 7190, canonical name is VietasFormula.
Accessed 2426 times total.

Classification:
AMS MSC12Y05 (Field theory and polynomials :: Computational aspects of field theory and polynomials)

Pending Errata and Addenda
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