To prove the division algorithm for polynomials using induction, we assume it holds for a polynomial of degree n and show it also holds for a polynomial of degree n+1. We divide the polynomial of degree n+1 by the divisor and use the inductive hypothesis to find the quotient and remainder. The remainder must be a polynomial of degree less than the divisor, which proves the base case. The inductive step involves showing that if the division algorithm holds for a polynomial of degree n, it also holds for a polynomial of degree n+1. We apply the inductive hypothesis to the quotient obtained from the division and show that the remainder is unique and has a degree less than the divisor, completing the proof.
