Vieta's formulas

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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 P ( x ) = a 0 x n + a 1 x n − 1 + ⋯ + a n − 1 x + a n {\displaystyle P(x)=a_{0}x^{n}+a_{1}x^{n-1}+\cdots +a_{n-1}x+a_{n}} {\displaystyle P(x)=a_{0}x^{n}+a_{1}x^{n-1}+\cdots +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, ..., rn by the fundamental theorem of algebra. Vieta's formulas relate the polynomial coefficients to signed sums of products of the roots r1, r2, ..., rn as follows:

{ r 1 + r 2 + ⋯ + r n − 1 + r n = − a 1 a 0 ( r 1 r 2 + r 1 r 3 + ⋯ + r 1 r n ) + ( r 2 r 3 + r 2 r 4 + ⋯ + r 2 r n ) + ⋯ + r n − 1 r n = a 2 a 0 ⋮ E k ( n ) ( r 1 , … , r n ) = ( − 1 ) k a k a 0 ⋮ r 1 r 2 ⋯ r n = ( − 1 ) n a n a 0 , {\displaystyle {\begin{cases}r_{1}+r_{2}+\dots +r_{n-1}+r_{n}=-{\dfrac {a_{1}}{a_{0}}}\\[1ex](r_{1}r_{2}+r_{1}r_{3}+\cdots +r_{1}r_{n})+(r_{2}r_{3}+r_{2}r_{4}+\cdots +r_{2}r_{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}}} {\displaystyle {\begin{cases}r_{1}+r_{2}+\dots +r_{n-1}+r_{n}=-{\dfrac {a_{1}}{a_{0}}}\\[1ex](r_{1}r_{2}+r_{1}r_{3}+\cdots +r_{1}r_{n})+(r_{2}r_{3}+r_{2}r_{4}+\cdots +r_{2}r_{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}}} *

where E k ( n ) ( r 1 , … , r n ) {\displaystyle E_{k}^{(n)}(r_{1},\ldots ,r_{n})} {\displaystyle E_{k}^{(n)}(r_{1},\ldots ,r_{n})} is the ⁠ k {\displaystyle k} {\displaystyle k}⁠th elementary symmetric polynomial in ⁠ n {\displaystyle n} {\displaystyle n}⁠ indeterminates; that is, E k ( n ) ( r 1 , … , r n ) = ∑ 1 ≤ i 1 < i 2 < ⋯ < i k ≤ n ( ∏ j = 1 k r i j ) {\displaystyle E_{k}^{(n)}(r_{1},\ldots ,r_{n})=\sum _{1\leq i_{1}<i_{2}<\cdots <i_{k}\leq n}\left(\prod _{j=1}^{k}r_{i_{j}}\right)} {\displaystyle E_{k}^{(n)}(r_{1},\ldots ,r_{n})=\sum _{1\leq i_{1}<i_{2}<\cdots <i_{k}\leq n}\left(\prod _{j=1}^{k}r_{i_{j}}\right)} is the sum of all products of ⁠ k {\displaystyle k} {\displaystyle k}⁠ ⁠ r i {\displaystyle r_{i}} {\displaystyle r_{i}}⁠ with different indices.

In particular, E 1 ( n ) ( r 1 , … , r n ) = r 1 + r 2 ⋯ + r n , E n ( n ) ( r 1 , … , r n ) = r 1 r 2 ⋯ r n , {\displaystyle {\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_{1}r_{2}\cdots r_{n},\end{aligned}}} {\displaystyle {\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_{1}r_{2}\cdots r_{n},\end{aligned}}} and the above expression of degree 2 in the ⁠ r i {\displaystyle r_{i}} {\displaystyle r_{i}}⁠ equals ⁠ E 2 ( n ) ( r 1 , … , r n ) {\displaystyle E_{2}^{(n)}(r_{1},\ldots ,r_{n})} {\displaystyle E_{2}^{(n)}(r_{1},\ldots ,r_{n})}⁠.

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 + ⋯ + a n {\displaystyle P(x)=a_{0}x^{n}+a_{1}x^{n-1}+\cdots +a_{n}} {\displaystyle P(x)=a_{0}x^{n}+a_{1}x^{n-1}+\cdots +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 ) ⋯ ( x − r n ) , {\displaystyle P(x)=a_{0}(x-r_{1})\cdots (x-r_{n}),} {\displaystyle P(x)=a_{0}(x-r_{1})\cdots (x-r_{n}),} where the ⁠ r i {\displaystyle r_{i}} {\displaystyle 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 + ⋯ + a n = a 0 ( x − r 1 ) ⋯ ( x − r n ) , {\displaystyle a_{0}x^{n}+a_{1}x^{n-1}+\cdots +a_{n}=a_{0}(x-r_{1})\cdots (x-r_{n}),} {\displaystyle a_{0}x^{n}+a_{1}x^{n-1}+\cdots +a_{n}=a_{0}(x-r_{1})\cdots (x-r_{n}),} then one has a i = ( − 1 ) i a 0 E i ( n ) ( r 1 , … , r n ) {\displaystyle a_{i}=(-1)^{i}a_{0}E_{i}^{(n)}(r_{1},\ldots ,r_{n})} {\displaystyle a_{i}=(-1)^{i}a_{0}E_{i}^{(n)}(r_{1},\ldots ,r_{n})} for ⁠ i = 1 , … , n {\displaystyle i=1,\ldots ,n} {\displaystyle i=1,\ldots ,n}⁠, where ⁠ E i ( n ) {\displaystyle E_{i}^{(n)}} {\displaystyle E_{i}^{(n)}}⁠ is the ⁠ i {\displaystyle i} {\displaystyle i}⁠th elementary symmetric polynomial, that is the sum of all products of ⁠ i {\displaystyle i} {\displaystyle i}⁠ of the ⁠ r k {\displaystyle r_{k}} {\displaystyle 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 {\displaystyle P(x)=x^{2}-1} {\displaystyle 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 ) {\displaystyle x^{2}-1=(x-1)(x-7)=(x-3)(x-5)} {\displaystyle x^{2}-1=(x-1)(x-7)=(x-3)(x-5)} As P ( x ) ≠ ( x − 1 ) ( x − 3 ) {\displaystyle P(x)\neq (x-1)(x-3)} {\displaystyle 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 {\displaystyle r_{1},r_{2}} {\displaystyle r_{1},r_{2}} of the quadratic polynomial P ( x ) = a x 2 + b x + c {\displaystyle P(x)=ax^{2}+bx+c} {\displaystyle P(x)=ax^{2}+bx+c} satisfy r 1 + r 2 = − b a , r 1 r 2 = c a . {\displaystyle r_{1}+r_{2}=-{\frac {b}{a}},\quad r_{1}r_{2}={\frac {c}{a}}.} {\displaystyle r_{1}+r_{2}=-{\frac {b}{a}},\quad 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 {\displaystyle r_{1},r_{2},r_{3}} {\displaystyle r_{1},r_{2},r_{3}} of the cubic polynomial P ( x ) = a x 3 + b x 2 + c x + d {\displaystyle P(x)=ax^{3}+bx^{2}+cx+d} {\displaystyle P(x)=ax^{3}+bx^{2}+cx+d} satisfy r 1 + r 2 + r 3 = − b a , r 1 r 2 + r 1 r 3 + r 2 r 3 = c a , r 1 r 2 r 3 = − d a . {\displaystyle r_{1}+r_{2}+r_{3}=-{\frac {b}{a}},\quad r_{1}r_{2}+r_{1}r_{3}+r_{2}r_{3}={\frac {c}{a}},\quad r_{1}r_{2}r_{3}=-{\frac {d}{a}}.} {\displaystyle r_{1}+r_{2}+r_{3}=-{\frac {b}{a}},\quad r_{1}r_{2}+r_{1}r_{3}+r_{2}r_{3}={\frac {c}{a}},\quad 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 + ⋯ + a 1 x + a 0 = a n ( x − r 1 ) ( x − r 2 ) ⋯ ( x − r n ) {\displaystyle 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})} {\displaystyle 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})} (which is true since r 1 , r 2 , … , r n {\displaystyle r_{1},r_{2},\dots ,r_{n}} {\displaystyle 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 {\displaystyle x} {\displaystyle x} between the two members of the equation.

Formally, if one expands ( x − r 1 ) ( x − r 2 ) ⋯ ( x − r n ) {\displaystyle (x-r_{1})(x-r_{2})\cdots (x-r_{n})} {\displaystyle (x-r_{1})(x-r_{2})\cdots (x-r_{n})} and regroup the terms by their degree in ⁠ x {\displaystyle x} {\displaystyle x}⁠, one gets

∑ k = 0 n ( − 1 ) n − k x k ( ∑ b 1 + ⋯ + b n = n − k ( ∀ i ) b i ∈ { 0 , 1 } r 1 b 1 ⋯ r n b n ) , {\displaystyle \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),} {\displaystyle \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),}

where the inner sum is exactly the ⁠ k {\displaystyle k} {\displaystyle 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 ) . {\displaystyle 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}).} {\displaystyle 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 {\displaystyle x} {\displaystyle x}, we find a 2 = a 2 {\displaystyle a_{2}=a_{2}} {\displaystyle a_{2}=a_{2}}, a 1 = − a 2 ( r 1 + r 2 ) {\displaystyle a_{1}=-a_{2}(r_{1}+r_{2})} {\displaystyle a_{1}=-a_{2}(r_{1}+r_{2})} and a 0 = a 2 ( r 1 r 2 ) {\displaystyle a_{0}=a_{2}(r_{1}r_{2})} {\displaystyle a_{0}=a_{2}(r_{1}r_{2})}, with which we can for example identify r 1 + r 2 = − a 1 / a 2 {\displaystyle r_{1}+r_{2}=-a_{1}/a_{2}} {\displaystyle r_{1}+r_{2}=-a_{1}/a_{2}} and r 1 r 2 = a 0 / a 2 {\displaystyle r_{1}r_{2}=a_{0}/a_{2}} {\displaystyle r_{1}r_{2}=a_{0}/a_{2}}, which are Vieta's formula's for n = 2 {\displaystyle n=2} {\displaystyle 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.[1] 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.[2][3]

In the opinion of the 18th-century British mathematician Charles Hutton, as quoted by Funkhouser,[4] 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.

Generalization to reflection groups

The permutation symmetry of the elementary symmetric polynomials can be considered as the Type An-1 case of a finite reflection group. The Chevalley–Shephard–Todd theorem gives analogs of the elementary symmetric polynomials for every such symmetry type, and Vieta's formulas generalize to give the coordinates of the quotient map of linear space by the reflection group.

See also

Notes

  1. R Rashed, Résolution des équations numériques et algèbre : Saraf-al-Din al-Tusi, Viète (French), Arch. History Exact Sci. 12 (1974), 244-290.
  2. Ypma, Tjalling J. (1995). "Historical Development of the Newton-Raphson Method". SIAM Review. 37 (4): 534. doi:10.1137/1037125. ISSN 0036-1445. JSTOR 2132904.
  3. "François Viète - Biography". Maths History. Retrieved 2025-01-07.
  4. (Funkhouser 1930)

References