Remainder and Factor Theorems

When one polynomial is divided by another you get a quotient and a remainder, just like dividing whole numbers. The Remainder Theorem is a shortcut for the special case where the divisor is linear, of the form (x − a). It says: the remainder is simply f(a). You do not need to carry out the long division at all. For example, to find the remainder when f(x) = x cubed minus 2x plus 5 is divided by (x − 1), just compute f(1) = 1 − 2 + 5 = 4. The remainder is 4. This works because dividing gives f(x) = (x − a)·q(x) + r, and substituting x = a makes (x − a) zero, leaving r = f(a).

The Factor Theorem is the Remainder Theorem with the remainder equal to zero. If f(a) = 0, then the remainder on dividing by (x − a) is zero, which means (x − a) divides f(x) exactly, so (x − a) is a factor. The statement works both ways: (x − a) is a factor of f(x) if and only if f(a) = 0. This is the test we use to hunt for factors of a polynomial: try small values like x = 1, −1, 2, −2 and see which make f(x) equal zero. Each value that gives zero hands us a factor.

The divisor is not always (x − a); it can be something like (2x + 3) or (3x − 1). To use the theorems, set the divisor equal to zero and solve for x. For (ax + b) = 0 we get x = −b/a, so the remainder is f(−b/a), and (ax + b) is a factor when f(−b/a) = 0. For example, to test whether (2x − 1) is a factor of f(x), set 2x − 1 = 0 giving x = one half, then check f(one half). If it is zero, (2x − 1) is a factor.

The biggest payoff is factorising a cubic, which long division alone makes painful. First use the Factor Theorem to find one factor: try x = 1, −1, 2, −2 until f(x) = 0. Suppose f(2) = 0, so (x − 2) is a factor. Then divide f(x) by (x − 2) using long division (or compare coefficients) to get a quadratic quotient. Finally factorise that quadratic by the usual methods. The cubic is now written as a product of three linear factors. This three-step routine, find a root, divide out, factorise the quadratic, solves almost every cubic in the ICSE syllabus.