Vieta’s formula

Suppose P(x) is a polynomialMathworldPlanetmathPlanetmathPlanetmath of degree n with roots r1,r2,,rn (not necessarily distinct). For 1kn, define Sk by


For example,


Then writing P(x) as


we find that


For example, if P(x) is a polynomial of degree 1, then P(x)=a1x+a0 and clearly r1=-a0a1.

If P(x) is a polynomial of degree 2, then P(x)=a2x2+a1x+a0 and r1+r2=-a1a2 and r1r2=a0a2. Notice that both of these formulas can be determined from the quadratic formula.

More intrestingly, if P(x)=a3x3+a2x2+a1x+a0, then r1+r2+r3=-a2a3, r1r2+r2r3+r3r1=a1a3, and r1r2r3=-a0a3.

Title Vieta’s formula
Canonical name VietasFormula
Date of creation 2013-03-22 15:21:55
Last modified on 2013-03-22 15:21:55
Owner neapol1s (9480)
Last modified by neapol1s (9480)
Numerical id 9
Author neapol1s (9480)
Entry type Theorem
Classification msc 12Y05
Related topic PropertiesOfQuadraticEquation