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.

Contents

[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, and an is nonzero), 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

\begin{cases} x_1 + x_2 + \dots + x_{n-1} + x_n = \tfrac{-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} \\ \vdots \\ x_1 x_2 \dots x_n = (-1)^n \tfrac{a_0}{a_n}. \end{cases}

In more detail, the sum of the products of all distinct of subsets k roots of P(X) equals ( − 1)kan − k / an, in other words (writing products by increasing indices to assure each subset of roots is used exactly once):

\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 integral domain, as long as the leading coefficient an is invertible (so that the divisions make sense) and the polynomial has n distinct roots in that ring. The condition of being an integral domain is needed to assure that a polynomial of degree n cannot have more than n roots, and that is if has n roots then it is determined (up to a scalar) by those roots. However, if one replaces the assumption that x1, …, xn are the roots of the polynomial by the simpler requirement that one has the relation

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),

then Viète's formulas even hold in any commutative ring, and they merely express the way the coefficients on the left hand side are formed when expanding the product on the left. Note that the given relation has to be required even in the case of an integral domain, if one wishes to allow some of the roots to coincide, and therefore needs to specify what multiplicity should be associated to each root.

[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 proven 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.