Factors of polynomials, the factor and remainder theorems

Learning objectives

Apply the remainder theorem to find the remainder when a polynomial is divided by (x − a) or (ax − b) by evaluating the polynomial at the appropriate value

Apply the factor theorem to test whether a given linear expression is a factor, and to find an unknown coefficient when a factor or remainder is specified

Factorise a cubic or quartic polynomial completely by identifying one factor with the factor theorem and then dividing or comparing coefficients to obtain the remaining quadratic

Solve a cubic equation by factorising it fully and solving each factor, stating all real roots

Set up and solve simultaneous equations in unknown coefficients from two pieces of information about factors or remainders