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 identical: divide the leading term, multiply, subtract, and repeat.

When This Method Applies

Use polynomial long division when factoring is not obvious, when you need the quotient and remainder directly, or when you want to rewrite an improper rational expression. It is also the required first move before partial fraction decomposition: if the numerator degree is at least as large as the denominator degree, you divide first. The stopping rule is simple — stop when the remainder has lower degree than the divisor. If the remainder is $0$, the division comes out evenly.

The whole reason the method works: at each stage you pick the quotient term that cancels the current leading term of the dividend, which is why the first move is always

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

Multiply the divisor by that term, subtract, and you are left with a new, smaller polynomial to continue with.

The 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 is far easier to get wrong.

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

We want

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

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

Round 1 — divide the leading terms

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

So the first quotient term is $2x^2$. Multiply by the divisor:

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

Subtract from the dividend:

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

Round 2 — repeat with the new leading term

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

Write $-x$ in the quotient. Multiply:

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

Subtract:

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

Round 3 — one more time

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

Write $3$ in the quotient. Multiply:

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

Subtract:

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

The remainder is $0$, so

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

Self-Check at Each Stage

The fastest verification is to multiply the quotient by the divisor:

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

which matches the original dividend, confirming the division.

The most common place to get stuck is a missing power. If you divide $x^3 + 4x - 1$ by $x - 1$, rewrite the dividend as

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

That $0x^2$ placeholder keeps every subtraction lined up. Without it, later terms slide into the wrong column — so if your columns ever look ragged, check for a skipped power before anything else.

Try It Yourself

Work through

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

Line up the degrees, then check by multiplication. For a harder follow-up, pick a case with a nonzero remainder and write the answer as

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