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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

 s_i=(-1)^i(a_(n-i))/(a_n).
(5)

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=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=sum_(i=1)^(3)r_i=-(a_2)/(a_3)
(16)
s_2=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=sum_(i,j,k; i<j<k)^(3)r_ir_jr_k
(20)
=r_1r_2r_3
(21)
=-(a_0)/(a_3).
(22)

See also

Newton-Girard Formulas, Polynomial Discriminant, Polynomial Roots, Symmetric Polynomial

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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 Formulas

Cite this as:

Weisstein, Eric W. "Vieta's Formulas." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/VietasFormulas.html

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