Reference no: EM131416904

We will now prove a fundamental result about polynomials: every non-zero polynomial of degreen (over a field F) has at most n roots. If you don't know what a field is, you can assume in thefollowing that F = R (the real numbers).

(a) Show that for any α ∈ F, there exists some polynomial Q(x) of degree n-1 and some b ∈ F such that P(x) = (x-α)Q(x) +b.

(b) Show that if α is a root of P(x), then P(x) = (x-α)Q(x).

(c) Prove that any polynomial of degree 1 has at most one root. This is your base case.(d) Now prove the inductive step: if every polynomial of degree n-1 has at most n-1 roots, thenany polynomial of degree n has at most n roots