What Is the Remainder Theorem? Formula With Examples

The Remainder Theorem is one of the most useful shortcuts in algebra. Instead of doing long polynomial division just to find a remainder, it lets you get the answer with a single substitution.

The Remainder Theorem formula

When a polynomial P(x) is divided by (x − a), the remainder is P(a).

That's it. You don't divide at all — you just plug a into the polynomial.

Remainder of P(x) ÷ (x − a) = P(a)

Why it works

Any polynomial division can be written as:

P(x) = (x − a) · Q(x) + R

where Q(x) is the quotient and R is the remainder (a constant, because the divisor is degree 1). If you substitute x = a, the (x − a) term becomes zero, leaving:

P(a) = 0 · Q(a) + R = R

So the remainder R is exactly P(a).

Worked example 1

Find the remainder when P(x) = x³ − 4x² + 5x − 2 is divided by (x − 3).

Here a = 3, so substitute:

P(3) = 3³ − 4(3²) + 5(3) − 2
     = 27 − 36 + 15 − 2
     = 4

The remainder is 4 — no long division needed. You can check the arithmetic with our scientific calculator.

Worked example 2 (divisor like x + 2)

Find the remainder when P(x) = 2x³ + x − 7 is divided by (x + 2).

Rewrite (x + 2) as (x − (−2)), so a = −2:

P(−2) = 2(−2)³ + (−2) − 7
      = 2(−8) − 2 − 7
      = −16 − 9 = −25

The remainder is −25.

The Factor Theorem connection

If the remainder P(a) comes out to zero, then (x − a) divides P(x) exactly — it's a factor. This is the Factor Theorem, and it's the basis for finding the roots of polynomials. For example, if P(1) = 0, then (x − 1) is a factor.

This links directly to solving equations: once you know a factor, you can reduce a cubic to a quadratic and finish with the quadratic equation calculator.

Quick recap

• Remainder of P(x) ÷ (x − a) is P(a).
• For (x + a), use a negative.
• If P(a) = 0, then (x − a) is a factor (Factor Theorem).

It turns a page of long division into one clean substitution.