The Remainder & Factor Theorems

Q: How does the Factor Theorem work?

If $p(a) = 0$, the remainder is zero, so $x - a$ divides evenly and is a factor — and $x = a$ is a root.

The Remainder Theorem gives a shortcut: to find the remainder when $p(x)$ is divided by $x - a$ , just evaluate $p(a)$ — no long division needed.

The Factor Theorem is the special case where that remainder is zero: if $p(a) = 0$ , then $x - a$ is a factor of $p(x)$ . Together they connect roots and factors.

The Remainder Theorem

Dividing $p(x)$ by $x - a$ leaves a remainder equal to $p(a)$ . So instead of doing the division, substitute $a$ into the polynomial.

For $p(x) = x^3 + 2x^2 - 5x + 1$ divided by $x - 2$ , the remainder is $p(2)$ .

The Factor Theorem

If $p(a) = 0$ , the remainder is zero, which means $x - a$ divides evenly — it is a factor. And every factor $x - a$ corresponds to a root $x = a$ .

This turns root-finding and factoring into the same job: test values, and any that give zero are both roots and factors.

Worked examples

Example 1: remainder by substitution

Find the remainder when $p(x) = x^3 + 2x^2 - 5x + 1$ is divided by $x - 2$ .

Evaluate p(2)

2^3 + 2(2^2) - 5(2) + 1

Simplify

8 + 8 - 10 + 1

Add

7

Answer: $7$

Example 2: testing a factor

Is $x - 1$ a factor of $p(x) = x^3 - 1$ ?

Check p(1)

1^3 - 1 = 0

Zero remainder means factor

\text{yes}

Answer: Yes, $x - 1$ is a factor

Try one yourself

Common questions

What does the Remainder Theorem save me?

The long division. To get the remainder from dividing by $x - a$ , just compute $p(a)$ .

How does the Factor Theorem work?

If $p(a) = 0$ , the remainder is zero, so $x - a$ divides evenly and is a factor — and $x = a$ is a root.

Are roots and factors the same thing?

They correspond one-to-one: the factor $x - a$ matches the root $x = a$ . Finding one gives the other.

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The Remainder & Factor Theorems

The Remainder Theorem

The Factor Theorem

Worked examples

Example 1: remainder by substitution

Example 2: testing a factor

Try one yourself

Common questions

Keep learning

Dividing Polynomials: Long & Synthetic Division

Roots & Zeros

Solving Polynomial Equations Algebraically