Vieta's formulas · Lablypedia

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Basic formulas

Any general polynomial of degree n $P (x) = a_{0} x^{n} + a_{1} x^{n - 1} + \dots + a_{n - 1} x + a_{n}$ (with the coefficients being real or complex numbers and a₀ ≠ 0) has n (not necessarily distinct) complex roots r₁, r₂, ..., r_n by the fundamental theorem of algebra. Vieta's formulas relate the polynomial coefficients to signed sums of products of the roots r₁, r₂, ..., r_n as follows:

where $E_{k}^{(n)} (r_{1}, \dots, r_{n})$ is the ⁠ $k$ ⁠th elementary symmetric polynomial in ⁠ $n$ ⁠ indeterminates; that is, $E_{k}^{(n)} (r_{1}, \dots, r_{n}) = \sum_{1 \leq i_{1} < i_{2} < \dots < i_{k} \leq n} (\prod_{j = 1}^{k} r_{i_{j}})$ is the sum of all products of ⁠ $k$ ⁠ ⁠ $r_{i}$ ⁠ with different indices.

In particular, $\begin{matrix} E_{1}^{(n)} (r_{1}, \dots, r_{n}) & = r_{1} + r_{2} \dots + r_{n}, \\ E_{n}^{(n)} (r_{1}, \dots, r_{n}) & = r_{1} r_{2} \dots r_{n}, \end{matrix}$ and the above expression of degree 2 in the ⁠ $r_{i}$ ⁠ equals ⁠ $E_{2}^{(n)} (r_{1}, \dots, r_{n})$ ⁠.

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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 $P (x) = a_{0} x^{n} + a_{1} x^{n - 1} + \dots + a_{n}$ of degree n over an integral domain R factors uniquely, up to the order of the factors, as $P (x) = a_{0} (x - r_{1}) \dots (x - r_{n}),$ where the ⁠ $r_{i}$ ⁠ belong to an algebraically closed field containing R.
If a polynomial of degree n factors over a commutative ring as $a_{0} x^{n} + a_{1} x^{n - 1} + \dots + a_{n} = a_{0} (x - r_{1}) \dots (x - r_{n}),$ then one has $a_{i} = (- 1)^{i} a_{0} E_{i}^{(n)} (r_{1}, \dots, r_{n})$ for ⁠ $i = 1, \dots, n$ ⁠, where ⁠ $E_{i}^{(n)}$ ⁠ is the ⁠ $i$ ⁠th elementary symmetric polynomial, that is the sum of all products of ⁠ $i$ ⁠ of the ⁠ $r_{k}$ ⁠ 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 $P (x) = x^{2} - 1$ has four roots (1, 3, 5, and 7) and two different factorizations: $x^{2} - 1 = (x - 1) (x - 7) = (x - 3) (x - 5)$ As $P (x) \neq (x - 1) (x - 3)$ , one cannot apply Vieta's formulas with the roots 1 and 3.

Example

Vieta's formulas applied to quadratic and cubic polynomials:

The roots $r_{1}, r_{2}$ of the quadratic polynomial $P (x) = a x^{2} + b x + c$ satisfy $r_{1} + r_{2} = - \frac{b}{a}, r_{1} r_{2} = \frac{c}{a} .$

The first of these equations can be used to find the minimum (or maximum) of P; see Quadratic equation § Vieta's formulas.

The roots $r_{1}, r_{2}, r_{3}$ of the cubic polynomial $P (x) = a x^{3} + b x^{2} + c x + d$ satisfy $r_{1} + r_{2} + r_{3} = - \frac{b}{a}, r_{1} r_{2} + r_{1} r_{3} + r_{2} r_{3} = \frac{c}{a}, r_{1} r_{2} r_{3} = - \frac{d}{a} .$

Proof

Vieta's formulas can be proved by considering the equality $a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{1} x + a_{0} = a_{n} (x - r_{1}) (x - r_{2}) \dots (x - r_{n})$ (which is true since $r_{1}, r_{2}, \dots, r_{n}$ are all the roots of this polynomial), expanding the products in the right-hand side, and equating the coefficients of each power of $x$ between the two members of the equation.

Formally, if one expands $(x - r_{1}) (x - r_{2}) \dots (x - r_{n})$ and regroup the terms by their degree in ⁠ $x$ ⁠, one gets

\sum_{k = 0}^{n} (- 1)^{n - k} x^{k} (\sum_{\overset{(\forall i) b_{i} \in {0, 1}}{b_{1} + \dots + b_{n} = n - k}} r_{1}^{b_{1}} \dots r_{n}^{b_{n}}),

where the inner sum is exactly the ⁠ $k$ ⁠th elementary symmetric function

As an example, consider the quadratic $f (x) = a_{2} x^{2} + a_{1} x + a_{0} = a_{2} (x - r_{1}) (x - r_{2}) = a_{2} (x^{2} - x (r_{1} + r_{2}) + r_{1} r_{2}) .$

Comparing identical powers of $x$ , we find $a_{2} = a_{2}$ , $a_{1} = - a_{2} (r_{1} + r_{2})$ and $a_{0} = a_{2} (r_{1} r_{2})$ , with which we can for example identify $r_{1} + r_{2} = - a_{1} / a_{2}$ and $r_{1} r_{2} = a_{0} / a_{2}$ , which are Vieta's formula's for $n = 2$ .

History

The formulas are named after the 16th-century French mathematician François Viète (Franciscus Vieta), who derived them for the case of positive roots. However, the methods of Vieta 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:

... [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.

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