Factor and Remainder Theorem

From the factorisation of polynomials we can derive two important theorems below. Factor theorem Theorem 1 (factor theorem). If is a factor of , then . Proof. Let be a factor of , then where is another factor of . Substitute into : Q.E.D.Corollary. If is a factor of , then Proof is left as an exercise to the reader. Remainder theorem Theorem 2. If when divided by gives remainder , then Proof. When is divided by , we write as . Substitute into : Q.E.D. Corollary. If is divided by and gives remainder , then Proof is left as an exercise to the reader. Example. is a factor of . Find the value of . Let . Factorise: . If is a factor of , then and are also factors of . According to the factor theorem, .