Polynomials

From the Fundamental Theorem of Algebra we know that an n-th degree univariate polynomial has n complex roots.

An example for the case n=3, let x^3+b*x^2+c*x+d=0 have roots r1, r2 and r3. Then

x^3+b*x^2+c*x+d=(x-r1)*(x - r2)*(x - r3),

expanding the right side of this equation gives

x^3+b*x^2+c*x+d=x^3-(r1+r2+r3)*x^2+(r1*r2+r1*r3+r2*r3)*x-r1*r2*r3.

This shows that the coefficients of an n-th degree univariate polynomial are multivariate polynomials in its roots.

Because changing the order of the roots does not change the coefficients the polynomials

-(r1+r2+r3),

(r1*r2+r1*r3+r2*r3) and

-r1*r2*r3,

are symmetric, and are called s(1), s(2) and s(3), or the elementary symmetric functions for n=3.