The Remainder Theorem Made Simple

When you divide a polynomial P(x) by (x − a), the remainder is P(a). You don't need long division.

The rule in one line

Remainder when P(x) is divided by (x − a) = P(a).

That means: plug in x = a, simplify, and that number is the remainder.

Worked example

Take P(x) = x³ − 2x + 5 divided by (x − 2).

1. The divisor is (x − 2), so a = 2.
2. Put x = 2 into P: P(2) = 2³ − 2(2) + 5 = 8 − 4 + 5.
3. 8 − 4 + 5 = 9. The remainder is 9.

You can check pieces on the scientific calculator: 2^3, then combine.

Another quick check

If the remainder is 0, then (x − a) is a factor of P(x). That links the remainder theorem to factoring.

Why this beats long division for remainders

Long division still works, but if the question only asks for the remainder, evaluating P(a) is faster and has fewer places to drop a sign.

Solving related quadratics? Try the Quadratic Solver.