Remainder Theorem in One Step

Quick answer: The remainder on dividing a polynomial f(x) by x − a is f(a). The quotient term vanishes at x = a in f(x) = (x − a)q(x) + r.

Worked example

Find the remainder when 2x³ − 3x + 1 is divided by x − 2.

1. Set a = 2.
2. Evaluate f(2) = 2(2³) − 3(2) + 1.
3. This is 16 − 6 + 1 = 11.

Check the result

A constant remainder is expected when dividing by a degree-one polynomial. No full long division is needed to find it.

Common mistake to avoid

For divisor x + 2, substitute −2, not 2. For kx − b with k ≠ 0, substitute b/k.

Try it yourself

1. Find the remainder for x² + 3x + 5 divided by x + 1. 2. Find the remainder for x² − 1 divided by x − 1.

Answers and reasoning

1. f(−1) = 1 − 3 + 5 = 3. 2. f(1) = 0, so the division is exact.

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