When you divide a polynomial f(x) by a linear factor (x - a), you usually grind through polynomial division to find the quotient and remainder. The remainder theorem gives the remainder instantly — no division needed:

Why? Write the division as f(x) = (x - a)\,q(x) + r, where q(x) is the quotient and r the (constant) remainder. Substitute x = a: the (x - a) term becomes 0, so f(a) = r. The remainder is simply the value of the polynomial at x = a.

The factor theorem is the special case where that remainder is zero. If f(a) = 0, then dividing by (x - a) leaves nothing over — so (x - a) divides f(x) exactly:

This lets you test a possible factor without dividing at all — just evaluate f(a) and check whether it is zero. It is the key that unlocks solving by factorising for higher-degree polynomials.

See it built

Step through evaluating f(a) for a chosen quadratic and a chosen a. The value you land on is exactly the remainder — and when it hits zero, you have found a factor.

See it explained

Sal Khan introduces the polynomial remainder theorem and shows why f(a) is the remainder.