Viète's formulas

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For Viète's formula for computing π, see that article.

In mathematics, more specifically in algebra, Viète's formulas, named after François Viète, are formulas which relate the roots of a polynomial to its coefficients.

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[edit] The formulas

If

P(X)=a_nX^n + a_{n-1}X^{n-1} +\cdots + a_1 X+ a_0

is a polynomial of degree n\ge 1 with complex coefficients (so the numbers a_0, a_1, \dots, a_{n-1}, a_n are complex with a_n\ne 0), by the fundamental theorem of algebra P(X) has n (not necessarily distinct) complex roots x_1, x_2, \dots, x_n. Viète's formulas state that

x_1 + x_2 + \cdots + x_n = -\frac{a_{n-1}}{a_n}
(x_1 x_2 + x_1 x_3+\cdots + x_1x_n) + (x_2x_3+x_2x_4+\cdots + x_2x_n)+\cdots + x_{n-1}x_n = \frac{a_{n-2}}{a_n}
\cdots\,
x_1 x_2 \cdots x_n = (-1)^n\frac{a_0}{a_n}.

In other words, the sum of all possible products of k roots of P(X) (with the indices in each product in increasing order so that there are no repetitions) equals ( − 1)kan − k / an,

\sum_{1\le i_1 < i_2 < \cdots < i_k\le n} x_{i_1}x_{i_2}\cdots x_{i_k}=(-1)^k\frac{a_{n-k}}{a_n}

for each k=1, 2, \dots, n.

Viète's formulas hold more generally for polynomials with coefficients in any commutative ring, as long as that polynomial of degree n has n roots in that ring.

[edit] Example

For the second degree polynomial P(X) = aX2 + bX + c, Viète's formulas state that the solutions x1 and x2 of the equation P(X) = 0 satisfy

x_1 + x_2 = - \frac{b}{a}, \quad x_1 x_2 = \frac{c}{a}.

The first of these equations can be used to find the minimum (or maximum) of P. See second order polynomial.

[edit] Proof

Viète's formulas can be proved by writing the equality

a_nX^n + a_{n-1}X^{n-1} +\cdots + a_1 X+ a_0 = a_n(X-x_1)(X-x_2)\cdots (X-x_n)

(which is true since x_1, x_2, \dots, x_n are all the roots of this polynomial), multiplying through the factors on the right-hand side, and identifying the coefficients of each power of X.

[edit] See also

[edit] References

  • Vinberg, E. B. (2003). A course in algebra. American Mathematical Society, Providence, R.I. ISBN 0821834134. 
  • Djukić, Dušan, et al. (2006). The IMO compendium: a collection of problems suggested for the International Mathematical Olympiads, 1959-2004. Springer, New York, NY. ISBN 0387242996. en:Viète's formulas
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