find coefficients of a polynomial given k roots

Question

Lets say that I have k roots for a polynomial and I am trying to find the coefficients of the terms in the polynomial.

(x - r1)(x - r2)(x - r3) ... (x - rk)

then I want to try to find the polynomials of the terms of x in the polynomial such that

a0 + a1 * x + a2 * x^2 + .... + ak *x^k

where a0 is the first coefficient (of x^0) and a^k is the coefficient of x^k (etc...) where each of those a terms will be defined in terms of roots.

So once again, my question is how do I solve for these coefficients in terms of the roots?

thanks

Side note: Ultimately I am trying to prove using induction that a given set of code will calculate these coefficients correctly, but I think that I can do that my self once I fully understand how the above work.

Answer 1

You just do the multiplication. This also is equivalent to the following convolutions: [1 -r1] * [1 -r2] * ... * [1 -rk] where * denotes convolution (which will give ak,...,a0)