How do you do polynomial long division step by step?

Four steps, repeated: divide the leading term of what is left by the leading term of the divisor, multiply the whole divisor by that result, subtract it, then bring down the next term. Stop when the degree of what remains is smaller than the degree of the divisor — that leftover is the remainder.

Why do I need placeholder terms for missing powers?

Long division lines up like powers in columns, and a missing power breaks the alignment. Dividing 2x³ + 3x² - 5 means writing it as 2x³ + 3x² + 0x - 5 first — without the 0x the -5 sits in the wrong column and every subtraction after that point is wrong.

What does it mean when the remainder is zero?

It means the divisor divides exactly, so it is a factor of the polynomial. If x - 1 divides x³ - 4x² + 5x - 2 with remainder 0, then x³ - 4x² + 5x - 2 = (x - 1)(x² - 3x + 2). This is the idea the Factor Theorem is built on.

How to Do Polynomial Division (Step by Step)

Dividing one polynomial by another is how you break a high-degree expression into smaller pieces you can actually work with. It is the tool behind factoring cubics and quartics, behind finding the zeros that a quadratic formula cannot reach, and behind rewriting an awkward rational expression into something you can integrate later in calculus. The good news is that you already know the algorithm — it is ordinary long division with in place of place value.

Polynomial Long Division — key formula — Key formula

What Polynomial Division Produces

Every division of polynomials produces the same four named pieces:

Symbol	Name	What it is
	dividend	the polynomial being divided
	divisor	the polynomial you are dividing by
	quotient	the result of the division
	remainder	what is left over, too small to divide further

This is the division algorithm for polynomials, and it is the same statement as for whole numbers. Two conditions come with it:

The remainder has a lower degree than the divisor . That is the stopping rule — you keep dividing until what is left is too small to divide further.
If , then is a factor of .

Dividing by a linear divisor like therefore always leaves a remainder that is a constant, because the only degree below is degree .

The Four Steps of Polynomial Long Division

Dividing polynomials using long division is a loop, not a formula. Each pass through the cycle removes one term from the dividend, and you keep looping until there is nothing left big enough to divide.

Step	What you do	Why
1. Divide	Divide the leading term of the current dividend by the leading term of the divisor	Gives the next term of the quotient
2. Multiply	Multiply the whole divisor by that new quotient term	Produces the piece you are about to remove
3. Subtract	Subtract that product from the current dividend	Cancels the leading term
4. Bring down	Bring down the next term of the original dividend	Sets up the next pass

Repeat until the degree of what remains drops below the degree of the divisor.

The divide, multiply, subtract, bring down cycle of polynomial long division — One pass removes one term; repeat until the remainder is too small to divide

The subtraction is where most errors happen. You are subtracting a whole expression, so every sign in it flips. Writing the product and then changing all its signs before adding is slower to write but far more reliable than subtracting in your head.

Polynomial Division Examples: Remainder Zero

The two worked examples below cover the only two outcomes there are — an exact division, and one that leaves something behind. Both are done step by step so you can follow the columns.

Divide by .

Step 1 — divide the leading terms:

Step 2 — multiply the divisor by and subtract:

Step 3 — bring down , then divide again:

Step 4 — bring down , then divide a final time:

The remainder is zero, so the division is exact:

And since , the full factorisation is — division turned a cubic you could not factor by inspection into two quadratic-sized problems.

Long Division Examples With Polynomials: A Missing Term

Most division of polynomials examples in a textbook are tidy. This one is not, and the untidiness is the lesson.

Divide by .

Step 1 — insert the placeholder. There is no term, so write one with coefficient zero:

Skipping this is the single most common way this problem goes wrong.

Step 2 — first pass:

Step 3 — bring down , second pass:

Step 4 — bring down , third pass:

The degree of is , below the divisor’s degree of , so we stop:

Long division tableau for 2x cubed plus 3x squared minus 5 divided by x plus 2 — The placeholder 0x keeps every power in its own column

Always check by multiplying back. Expanding gives ; subtracting returns . The check costs thirty seconds and catches every sign slip.

Common Mistakes to Avoid

Omitting placeholder terms. A dividend missing a power silently misaligns every column after it. Write , — whatever is absent — before you start.
Subtracting only the first term of the product. You are subtracting the entire expression, so every sign inside it flips. is , not .
Multiplying by only part of the divisor. Each quotient term multiplies the whole divisor, not just its leading term.
Stopping too early or too late. The stopping rule is degree, not appearance: keep going while the current remainder’s degree is at least the divisor’s, and stop the moment it drops below.
Forgetting the divisor’s own sign. Dividing by is not the same as dividing by , and the sign carries through every multiply step.

Practice Problems

Six long division of polynomials problems, working up from a clean quadratic to a check-and-diagnose puzzle. Try each one before opening the answer.

Problem 1. Divide by .

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Step 1 — divide leading terms and subtract:

Step 2 — bring down and repeat:

Answer: with remainder , so

Problem 2. Divide by .

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Step 1 — insert both placeholders:

Step 2 — first pass:

Step 3 — second pass:

Step 4 — third pass:

Answer: with remainder — the difference-of-cubes factorisation

Problem 3. Divide by .

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Step 1 — first pass:

Step 2 — bring down , second pass:

Answer: quotient , remainder , so

Problem 4. Divide by .

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Step 1 — write in the placeholders:

Step 2 — first pass:

Step 3 — bring down the rest and divide again:

Answer: with remainder . Note the divisor here has degree , so the process stops as soon as the remainder drops below degree .

Problem 5. A student divides by and gets quotient with remainder . Without redoing the division, show that this is wrong.

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Step 1 — multiply back and see whether it reproduces the dividend:

Step 2 — compare with the original. The original has no term; this has .

Answer: the check fails, so the division is wrong. The student almost certainly omitted the placeholder — which is exactly the error that leaves a stray term unaccounted for.

Problem 6. If divides a polynomial exactly, what is ?

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Step 1 — “divides exactly” means the remainder is zero, so:

Step 2 — substitute :

Answer: . This is the Factor Theorem in one line — an exact divisor always corresponds to a zero at .

Once you are comfortable with the full tableau, most divisions by a linear factor can be done far faster with synthetic division, which strips the same arithmetic down to a row of coefficients. Division also pairs naturally with end behavior when you are sketching: division finds the factors that fix the middle of the curve, end behavior fixes the two tails.

Polynomial Long Division

What Polynomial Division Produces

The Four Steps of Polynomial Long Division

Polynomial Division Examples: Remainder Zero

Long Division Examples With Polynomials: A Missing Term

Common Mistakes to Avoid

Practice Problems

Frequently Asked Questions

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