(A Level) Polynomial division, factor and remainder theorems

Learning objectives

Divide a polynomial of degree up to 4 by a linear or quadratic divisor, by long division or by comparing coefficients, and state the quotient and the remainder

Use the factor theorem to show that (x − a) or (ax + b) is a factor by showing that p(a) = 0 or p(−b/a) = 0

Use the remainder theorem to find the remainder when p(x) is divided by (ax + b), and form simultaneous equations for unknown coefficients from given factors and remainders

Factorise cubic and quartic polynomials completely and solve p(x) = 0, including equations that become polynomial after a substitution such as u = 2ˣ or u = tan θ