Vieta's Formulas
Let s_i be the sum of the products of distinct polynomial roots r_j of the polynomial equation of degree n
| a_nx^n+a_(n-1)x^(n-1)+...+a_1x+a_0=0, |
(1)
|
where the roots are taken i at a time (i.e., s_i is defined as the symmetric polynomial Pi_i(r_1,...,r_n)) s_i is defined for i=1, ..., n. For example, the first few values of s_i are
s_1 = r_1+r_2+r_3+r_4+...
(2)
s_2 = r_1r_2+r_1r_3+r_1r_4+r_2r_3+...
(3)
s_3 = r_1r_2r_3+r_1r_2r_4+r_2r_3r_4+...,
(4)
and so on. Then Vieta's formulas states that
The theorem was proved by Viète (also known as Vieta, 1579) for positive roots only, and the general theorem was proved by Girard.
This can be seen for a second-degree polynomial by multiplying out,
a_2x^2+a_1x+a_0 = a_2(x-r_1)(x-r_2)
(6)
= a_2[x^2-(r_1+r_2)x+r_1r_2],
(7)
so
s_1 = sum_(i=1)^(2)r_i
(8)
= r_1+r_2
(9)
= -(a_1)/(a_2)
(10)
s_2 = [画像:sum_(i,j=1; i!=j)^(2)r_ir_j]
(11)
= r_1r_2
(12)
= (a_0)/(a_2).
(13)
Similarly, for a third-degree polynomial,
a_3x^3+a_2x^2+a_1x+a_0 = a_3(x-r_1)(x-r_2)(x-r_3)
(14)
= a_3[x^3-(r_1+r_2+r_3)x^2+(r_1r_2+r_1r_3+r_2r_3)x-r_1r_2r_3],
(15)
so
s_1 = [画像:sum_(i=1)^(3)r_i=-(a_2)/(a_3)]
(16)
s_2 = [画像:sum_(i,j; i<j)^(3)r_ir_j]
(17)
= r_1r_2+r_1r_3+r_2r_3
(18)
= (a_1)/(a_3)
(19)
s_3 = [画像:sum_(i,j,k; i<j<k)^(3)r_ir_jr_k]
(20)
= r_1r_2r_3
(21)
= [画像:-(a_0)/(a_3).]
(22)
See also
Newton-Girard Formulas, Polynomial Discriminant, Polynomial Roots, Symmetric PolynomialExplore with Wolfram|Alpha
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References
Bold, B. Famous Problems of Geometry and How to Solve Them. New York: Dover, p. 56, 1982.Borwein, P. and Erdélyi, T. "Newton's Identities." §1.1.E.2 in Polynomials and Polynomial Inequalities. New York: Springer-Verlag, pp. 5-6, 1995.Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, pp. 1-2, 1959.Girard, A. Invention nouvelle en l'algèbre. Leiden, Netherlands: Bierens de Haan, 1884.Hazewinkel, M. (Managing Ed.). Encyclopaedia of Mathematics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia," Vol. 9. Dordrecht, Netherlands: Reidel, p. 416, 1988.van der Waerden, B. L. Algebra, Vol. 1. New York: Springer-Verlag, 1993.Viète, F. Opera mathematica. 1579. Reprinted Leiden, Netherlands, 1646.Referenced on Wolfram|Alpha
Vieta's FormulasCite this as:
Weisstein, Eric W. "Vieta's Formulas." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/VietasFormulas.html