Vieta's formulas

thumb|[[François Viète]]

In mathematics, Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. They are named after François Viète (1540-1603), more commonly referred to by the Latinised form of his name, "Franciscus Vieta."

Basic formulas

Any general polynomial of degree n

<math display="block">P(x) = a_0 x^n + a_1 x^{n-1} + \cdots + a_{n-1} x + a_n</math>

(with the coefficients being real or complex numbers and ) has (not necessarily distinct) complex roots by the fundamental theorem of algebra. Vieta's formulas relate the polynomial coefficients to signed sums of products of the roots as follows:

{a_0} \\[1ex]

(r_1 r_2 + r_1 r_3 + \cdots + r_1 r_n) + (r_2r_3 + r_2r_4+\cdots + r_2r_n)+\cdots + r_{n-1}r_n = \dfrac{a_{2{a_{0 \\[1ex]

{} \quad \vdots \\[1ex]

E_k^{(n)}(r_1,\ldots,r_n) = (-1)^k\dfrac{a_k}{a_0} \\[1ex]

{} \quad \vdots \\[1ex]

r_1 r_2 \cdots r_n = (-1)^n \dfrac{a_n}{a_0},

\end{cases}</math>|

where <math >E_k^{(n)}(r_1,\ldots,r_n)</math> is the th elementary symmetric polynomial in indeterminates; that is,

<math display="block">E_k^{(n)}(r_1,\ldots,r_n)=\sum_{1\le i_1 < i_2 < \cdots < i_k\le n} \left(\prod_{j = 1}^k r_{i_j}\right)</math>

is the sum of all products of with different indices.

In particular,

<math display="block">\begin{aligned}

E_1^{(n)}(r_1,\ldots,r_n)&= r_1+r_2\cdots + r_n,\\

E_n^{(n)}(r_1,\ldots,r_n)&= r_1r_2\cdots r_n,

\end{aligned}

</math>

and the above expression of degree 2 in the equals .

Generalization to rings

As stated above, Vieta's formulas remain valid when the coefficients of the polynomial belong to an integral domain and the roots belong to an algebraically closed field. For a further generalization, it is useful to separate the statement in two parts:

Every polynomial <math display=block> P(x)=a_0x^n+a_1x^{n-1}+\cdots +a_n</math>of degree over an integral domain factors uniquely, up to the order of the factors, as <math display=block> P(x)=a_0(x-r_1)\cdots(x-r_n),</math> where the belong to an algebraically closed field containing .
If a polynomial of degree factors over a commutative ring as <math display=block> a_0x^n+a_1x^{n-1}+\cdots +a_n=a_0(x-r_1)\cdots(x-r_n),</math> then one has <math display=block> a_i= (-1)^ia_0 E_i^{(n)}(r_1,\ldots,r_n)</math> for , where is the th elementary symmetric polynomial, that is the sum of all products of of the with different indices.

Vieta's formulas are useful over an integral domain, because they provide relations between the roots without having to compute them.

Over a ring that is not an integral domain, some care is needed. For example, in the ring of the integers modulo 8, the quadratic polynomial <math>P(x) = x^2-1</math> has four roots (1, 3, 5, and 7) and two different factorizations:<math display=block>x^2-1=(x-1)(x-7)=(x-3)(x-5)</math> As <math>P(x)\neq (x-1)(x-3)</math>, one cannot apply Vieta's formulas with the roots and .

Example

Vieta's formulas applied to quadratic and cubic polynomials:

The roots <math>r_1, r_2</math> of the quadratic polynomial <math>P(x) = ax^2 + bx + c</math> satisfy

The first of these equations can be used to find the minimum (or maximum) of ; see .

The roots <math>r_1, r_2, r_3</math> of the cubic polynomial <math>P(x) = ax^3 + bx^2 + cx + d</math> satisfy

Proof

Vieta's formulas can be proved by considering the equality

<math display="block">a_n x^n + a_{n-1}x^{n-1} +\cdots + a_1 x+ a_0 = a_n (x-r_1) (x-r_2) \cdots (x-r_n)</math>

(which is true since <math>r_1, r_2, \dots, r_n</math> are all the roots of this polynomial), expanding the products in the right-hand side, and equating the coefficients of each power of <math>x</math> between the two members of the equation.

Formally, if one expands <math>(x-r_1) (x-r_2) \cdots (x-r_n)</math> and regroup the terms by their degree in , one gets

:<math>\sum_{k=0}^n (-1)^{n-k}x^k \left(\sum_{\stackrel{(\forall i)\; b_i\in\{0,1\{b_1+\cdots+b_n=n-k r_1^{b_1}\cdots r_n^{b_n}\right),</math>

where the inner sum is exactly the th elementary symmetric function

As an example, consider the quadratic

Comparing identical powers of <math>x</math>, we find <math>a_2=a_2</math>, <math>a_1=-a_2 (r_1+r_2) </math> and <math> a_0 = a_2 (r_1r_2) </math>, with which we can for example identify <math> r_1+r_2 = - a_1/a_2 </math> and <math> r_1r_2 = a_0/a_2 </math>, which are Vieta's formula's for <math>n=2</math>.

History

The formulas are named after the 16th-century French mathematician François Viète, who derived them for the case of positive roots. However, the methods of Viète and those of the 12th-century Islamic mathematician Sharaf al-Din al-Tusi were very close to each other. It is plausible that algebraic advancements made by other Islamic mathematicians such as Omar Khayyam, al-Tusi, and al-Kashi influenced 16th-century algebraists, with Vieta being the most prominent among them.

In the opinion of the 18th-century British mathematician Charles Hutton, as quoted by Funkhouser, the general principle (not restricted to positive real roots) was first understood by the 17th-century French mathematician Albert Girard:

<blockquote>...[Girard was] the first person who understood the general doctrine of the formation of the coefficients of the powers from the sum of the roots and their products. He was the first who discovered the rules for summing the powers of the roots of any equation.</blockquote>

Vieta's formulas

Basic formulas

Generalization to rings

Example

Proof

History

See also

Notes

References