The Remainder Theorem

The Remainder Theorem

When a polynomial $f(x)$ is divided by $(x - c)$, the remainder equals $f(c)$. This means you can find the remainder by simply plugging in $c$ — no long division needed!

Step-by-step: Find the remainder when $x^4 - 3x^3 + 2x - 5$ is divided by $(x - 2)$

Step 1 — Identify $c$. Dividing by $(x - 2)$ means $c = 2$.

Step 2 — Evaluate $f(2)$:

$f(2) = (2)^4 - 3(2)^3 + 2(2) - 5$ $= 16 - 3(8) + 4 - 5$ $= 16 - 24 + 4 - 5$ $= -9$

Answer: The remainder is $-9$.

Step 3 — Verify: If we did the full division, we'd get $f(x) = (x-2) \cdot Q(x) + (-9)$ for some quotient $Q(x)$.

Connection to the Factor Theorem

The Factor Theorem is a special case of the Remainder Theorem: if $f(c) = 0$ (remainder is zero), then $(x - c)$ divides evenly — it's a factor.

So: remainder = 0 → $(x - c)$ is a factor. Remainder $\neq$ 0 → not a factor.

Why it works

By the division algorithm: $f(x) = (x - c) \cdot Q(x) + R$. Plug in $x = c$: $f(c) = (c - c) \cdot Q(c) + R = 0 + R = R$.

Try it in the visualization

Adjust $c$ and the coefficients. The point $f(c)$ is computed step by step and marked on the graph. If $f(c) = 0$, the theorem confirms $(x - c)$ is a factor.