The Remainder and Factor Theorems

By the end of this lesson, you will be able to apply the Remainder Theorem to evaluate polynomials efficiently, use the Factor Theorem to test for factors without full division, and combine these tools to find zeros of higher-degree polynomials.

The Big Idea

Division gives us two theorems that eliminate much of the computation normally needed:

• Remainder Theorem: instead of dividing by $(x - r)$ and reading the remainder, just evaluate $p(r)$.
• Factor Theorem: $(x - r)$ is a factor iff $p(r) = 0$.

Together, these transform "is $r$ a zero?" from a division problem into a substitution.

The Remainder Theorem

When $p(x)$ is divided by $(x - r)$, the remainder equals $p(r)$.

$$p(x) = (x - r) \cdot q(x) + p(r)$$

Why: substituting $x = r$ into both sides: $p(r) = (r-r) \cdot q(r) + p(r) = p(r)$ ✓

The Factor Theorem

$(x - r)$ is a factor of $p(x)$ if and only if $p(r) = 0$.

This is just the Remainder Theorem with remainder $= 0$: zero remainder means $(x - r)$ divides evenly.

Example 1 — Remainder Theorem

Find the remainder when $p(x) = 3x^4 - 2x^2 + x - 5$ is divided by $(x - 2)$.

$p(2) = 3(16) - 2(4) + 2 - 5 = 48 - 8 + 2 - 5 = 37$

Remainder = 37 (no division required).

Example 2 — Factor Theorem

Is $(x - 3)$ a factor of $p(x) = 2x^3 - 5x^2 - 4x + 3$?

$p(3) = 2(27) - 5(9) - 12 + 3 = 54 - 45 - 12 + 3 = 0$ ✓

Yes — $(x - 3)$ is a factor.

Example 3 — Finding the Quotient After Confirming a Factor

$p(x) = 2x^3 - 5x^2 - 4x + 3$, factor $(x - 3)$ confirmed. Find the complete factorization.

Synthetic division by $r = 3$:

 3 | 2   -5   -4    3
   |      6    3   -3
   |-----------------
     2    1   -1    0

Quotient: $2x^2 + x - 1 = (2x - 1)(x + 1)$

Full factorization: $(x-3)(2x-1)(x+1)$

Zeros: $x = 3, \frac{1}{2}, -1$

Example 4 — Using Remainder Theorem for Evaluation

Evaluate $q(x) = x^5 - 3x^3 + 2x - 7$ at $x = -2$ using synthetic division.