The Factor Theorem and Remainder Theorem

The Factor Theorem and Remainder Theorem are powerful tools for working with polynomials. They allow you to test for factors, find roots, and factorise cubics and higher-degree polynomials efficiently. Both are explicitly required by the Edexcel 9MA0 specification.

The Remainder Theorem

Statement: If a polynomial f(x) is divided by (x − a), the remainder is f(a).

More generally, if f(x) is divided by (ax − b), the remainder is f(b/a).

Example: Find the remainder when f(x) = 2x³ − 5x² + 3x + 7 is divided by (x − 2).

f(2) = 2(8) − 5(4) + 3(2) + 7 = 16 − 20 + 6 + 7 = 9.

The remainder is 9.

Example: Find the remainder when f(x) = 3x³ + x − 4 is divided by (3x − 1).

Evaluate f(1/3) = 3(1/27) + 1/3 − 4 = 1/9 + 1/3 − 4 = 1/9 + 3/9 − 36/9 = −32/9.