Polynomial Long Division

Polynomial long division is a step-by-step way to divide one polynomial by another by hand. If you know number long division, the pattern is the same: divide the leading term, multiply, subtract, and repeat.

The key stopping rule is simple. Stop when the remainder has lower degree than the divisor. If the remainder is $0$, the division comes out evenly.

Why Polynomial Long Division Works

At each stage, you choose the quotient term that will cancel the current leading term of the dividend.

That is why the first move is always:

$$\frac{\text{leading term of dividend}}{\text{leading term of divisor}}$$

Once you have that quotient term, multiply the whole divisor by it and subtract. The subtraction creates a new, smaller polynomial to continue with.

Polynomial Long Division Steps

1. Write both polynomials in descending powers.
2. Insert missing powers with coefficient $0$ if needed.
3. Divide the leading term of the current dividend by the leading term of the divisor.
4. Write that result in the quotient.
5. Multiply the divisor by that quotient term.
6. Subtract.
7. Bring down the next term and repeat.

If the terms are not lined up by degree, the subtraction step becomes much easier to get wrong.

Worked Example: Divide $2x^3 - 5x^2 + 5x - 6$ by $x - 2$

We want to find

$$\frac{2x^3 - 5x^2 + 5x - 6}{x - 2}.$$

The goal in each round is to cancel the current leading term.

1. Divide the leading terms

Divide $2x^3$ by $x$:

$$\frac{2x^3}{x} = 2x^2.$$

So the first term of the quotient is $2x^2$.

2. Multiply and subtract

Multiply $2x^2$ by the divisor:

$$2x^2(x - 2) = 2x^3 - 4x^2.$$

Subtract from the original dividend:

$$(2x^3 - 5x^2 + 5x - 6) - (2x^3 - 4x^2) = -x^2 + 5x - 6.$$

3. Repeat with the new leading term

Now divide $-x^2$ by $x$:

$$\frac{-x^2}{x} = -x.$$

Write $-x$ in the quotient.

Multiply:

$$-x(x - 2) = -x^2 + 2x.$$

Subtract:

$$(-x^2 + 5x - 6) - (-x^2 + 2x) = 3x - 6.$$

4. One more round

Divide $3x$ by $x$:

$$\frac{3x}{x} = 3.$$

Write $3$ in the quotient.

Multiply:

$$3(x - 2) = 3x - 6.$$

Subtract:

$$(3x - 6) - (3x - 6) = 0.$$

So the remainder is $0$, and the quotient is

$$\frac{2x^3 - 5x^2 + 5x - 6}{x - 2} = 2x^2 - x + 3.$$

How To Check Your Answer

Multiply the quotient by the divisor:

$$(2x^2 - x + 3)(x - 2).$$

Expanding gives

$$2x^3 - 5x^2 + 5x - 6,$$

which matches the original dividend. That confirms the division is correct.

Common Mistake: Skipping a Missing Power

The most common setup error is skipping a missing power. For example, if you divide $x^3 + 4x - 1$ by $x - 1$, you should rewrite the dividend as

$$x^3 + 0x^2 + 4x - 1.$$

That $0x^2$ placeholder keeps every subtraction lined up. Without it, later terms can slide into the wrong column.

When You Use Polynomial Long Division

This method is useful when factoring is not obvious, when you need the quotient and remainder directly, or when you want to rewrite an improper rational expression.

It also appears before partial fraction decomposition. If the numerator degree is at least as large as the denominator degree, polynomial long division comes first.

Try One On Your Own

Try your own version with

$$\frac{x^3 + 2x^2 - 5x - 6}{x + 3}.$$

Focus on lining up degrees and checking the result by multiplication. For a useful next step, try a case with a nonzero remainder and write the answer as

$$\text{quotient} + \frac{\text{remainder}}{\text{divisor}}.$$