Expand Fundamental Theorem of Algebra

Question

The fundamental theorem of algebra tells us that polynomial equation of degree $n$ can be written as

$$ p(z) = a_0(z-z_1)(z-z_2)...(z-z_n) $$

How would one go about expanding this expression and what would the expansion look like?

Answer 1

If you look at small powered expressions, you can find the pattern nicely. For example,

$$(z-z_1)(z-z_2)=z^2-(z_1+z_2)z+z_1z_2$$

$$(z-z_1)(z-z_2)(z-z_3)=z^3-(z_1+z_2+z_3)z^2+(z_1z_2+z_2z_3+z_1z_3)z-z_1z_2z_3$$

Note the sum of the roots is the coefficient for the term of degree 1 less than that of the number of roots. Note the constant term is the product of the roots...and certainly if you multiply by a constant $a_0$, what would happen there?

Answer 2

The coefficients happen to be the elementary symmetric polynomials in the roots $z_1,\dots,z_n$.