Remainder theorem
A standard method from More about polynomials: the steps and worked examples.
Steps
When f(x) is divided by (x − a), the remainder is f(a). For (ax − b), the remainder is f(b/a).
- Find the value that makes the divisor zero: x − a gives a; 2x + 1 gives −1/2.
- Substitute that value into f(x).
- The result is the remainder; no long division is needed.
Why: f(x) = (x − a) × Q(x) + R. Put x = a and the first part becomes 0, leaving R.
Worked examples
Example 1: Remainder theorem: find the remainder when x³ − 2x² + 4x − 7 is divided by x − 2.
f(2) = 8 − 8 + 8 − 7 = 1. (Long division gives quotient x² + 4, remainder 1.)
Example 2: Factor theorem: factorise the cubic f(x) = x³ − 4x² + x + 6.
f(−1) = −1 − 4 − 1 + 6 = 0, so (x + 1) is a factor
f(x) ÷ (x + 1) = x² − 5x + 6
f(x) = (x + 1)(x − 2)(x − 3)For the whole topic, read More about polynomials: key points, common mistakes and how the DSE tests it.