How to prove this proposition with induction

Question

Let $P(x)$ be a polynomial of degree $n$ in the field $\mathbb{R}$ such that $a_n,\ldots,a_0$ are the coefficients. How can I show through induction that if there is at least one coefficient $a_i$ that is not $0$ then there are at most $n $ different roots to $P(x)$?

Answer 1

Try polynomial division for linear factors, for instance using the Newton-Lagrange-Horner-Ruffini scheme. The result is a polynomial of one degree less, enabling induction.