What Is the Remainder Theorem?

The rule

When a polynomial P(x) is divided by x − a, the remainder is P(a). Substitute a into the polynomial instead of doing a full division.

P(x) = x² + 3x + 2. Dividing by x − 2 gives a remainder of P(2) = 4 + 6 + 2 = 12.

Why it works

Polynomial division writes P(x) = (x − a)Q(x) + r. Setting x = a makes the first term zero, leaving P(a) = r.

The factor theorem

If P(a) = 0, the remainder is zero, so x − a is a factor of P(x).

For P(x) = x² − 9, P(3) = 0. Therefore x − 3 is a factor, and x² − 9 = (x − 3)(x + 3).