How to Do Polynomial Long Division, Step by Step

Most people assume polynomial long division is a different beast from the number long division they learned in grade school. It is not. If you can divide by 7, you already know the skeleton of how to do polynomial long division. The trouble starts when terms go missing and when a remainder shows up that is not a number. This method gives you the exact, step-by-step process to reproduce it on a test without freezing. You will get the four-step loop, two full worked examples in long-division format, and the one habit that separates students who pass from students who guess.

What Polynomial Long Division Is, Compared to Numbers

Polynomial long division is the algorithm for dividing one polynomial by another, producing a quotient and a remainder. The remainder is a polynomial, not a number, unless the divisor is linear. The relationship is always P(x) = D(x) * Q(x) + R(x), where P is the dividend, D is the divisor, Q is the quotient, and R is the remainder. An x-squared term is not ten times an x term; it is the variable squared, and the place value works by degree, not by factor of ten. If the dividend has no x-squared term, you put a zero placeholder there. Skipping it misaligns the subtraction and ruins the entire problem. That remainder is written as a fraction over the divisor, never as a decimal.

Setting Up the Division: Standard Form and Placeholders

Before you divide, write both polynomials in standard form, from highest degree to lowest. A polynomial like 3x^2 + 5 - 2x^3 must be rewritten as -2x^3 + 3x^2 + 5 before you start. If either polynomial has missing terms, insert a zero placeholder for each one. The placeholder is not optional. It holds the column open so that when you subtract, the like terms line up. Misalignment is the number one cause of wrong answers in polynomial long division, and it is almost always because a missing term was skipped.

The Four-Step Loop: Divide, Multiply, Subtract, Bring Down

Here is the heart of how to do polynomial long division. Each round of the algorithm has exactly four steps.

Divide: Take the leading term of the current dividend and divide it by the leading term of the divisor. Write the result above the long division bracket, aligned over the term of the same degree.
Multiply: Multiply the entire divisor by that quotient term. This changes the sign of each term in the product, so you add the opposite. Write the result below.
Bring Down: Bring down the next term from the original dividend. Each round uses the new current dividend, never the original one. The quotient accumulates above the bracket. When you cannot divide anymore, the final expression is written as Q(x) + R(x)/D(x), where Q is the quotient, R is the remainder, and D is the divisor.

Worked Example 1: Dividing x^2 + 5x + 6 by x + 2

Write x above the bracket, aligned over the x^2 term. Multiply x by x + 2, get x^2 + 2x, subtract, bring down the 5x, get 3x + 6. Divide 3x by x, get 3. Write +3 above the bracket, aligned over the constant term. Multiply 3 by x + 2, get 3x + 6, subtract, remainder 0. Check by multiplying: (x + 2)(x + 3) equals x^2 + 5x + 6.

For the second example, divide 2x^3 - 5x + 7 by x^2 + x. Write 2x above the bracket, aligned over the x^3 position. Multiply 2x by x^2 + x, get 2x^3 + 2x^2, subtract, bring down -5x, get -2x^2 - 5x. Divide -2x^2 by x^2, get -2. Write -2 above the bracket, aligned over the constant term. Multiply -2 by x^2 + x, get -2x^2 - 2x, subtract, bring down 7, get -3x + 7. The quotient is 2x - 2, and the remainder is -3x + 7.

The answer is written as 2x - 2 + (-3x + 7)/(x^2 + x). Check by multiplying the quotient by the divisor and adding the remainder: (x^2 + x)(2x - 2) equals 2x^3 - 2x^2 + 2x^2 - 2x, which simplifies to 2x^3 - 2x. Adding the remainder, -3x + 7, gives 2x^3 - 5x + 7, the original dividend. The remainder is a polynomial, not a number, because the divisor is degree 2. This is the part most students get wrong: they try to force a number remainder when the remainder is a polynomial that cannot be divided further.

Writing the Answer as Q(x) + R(x)/D(x)

When the remainder is not zero, the final answer always takes the form quotient plus remainder over divisor. The quotient is the expression above the bracket. The answer is 2x - 2 + (-3x + 7)/(x^2 + x).

This fraction is not a decimal and is not simplified into one. If you see a calculator output that shows decimals, it is rounding. The remainder has a specific role: it tells you how far off the dividend is from being a multiple of the divisor. In graphing rational functions, it determines the oblique asymptote when the degree of the numerator is exactly one more than the degree of the denominator.

Polynomial Long Division Steps: The Full Sequence in Brief

Here is the entire method compressed into a sequence you can drill before the test. First, write both polynomials in standard form. Second, insert zero placeholders for any missing terms in the dividend. Third, divide the leading term of the dividend by the leading term of the divisor. Fourth, multiply the entire divisor by that result and subtract. Fifth, bring down the next term and repeat. Sixth, write the answer as Q(x) + R(x)/D(x).

Memorize this sequence as a unit, not as separate ideas. On a test, you will not have time to reconstruct the logic from scratch. Drill it with five to ten problems before exam day. Use varied examples: one with a missing term, one with a non-monic divisor, one with a remainder. The method is the same every time. If you can do the arithmetic cleanly, you can divide any polynomial.

Polynomial Long Division with Missing Terms

When the dividend skips a term, such as x^3 - 4x + 7, the missing x^2 term is not optional. You must write 0x^2. This is the most common error in polynomial long division examples. Skipping the placeholder shifts every term one column to the left, and the subtraction step produces garbage.

Here is the rule: for every degree from the highest down to zero, if the term is absent, insert a zero. The benefit is that the columns stay straight, and you can see exactly which terms subtract from which. Do not skip it.

How to Divide Polynomials When the Divisor Is Not Monic

A monic polynomial has a leading coefficient of 1. When the divisor is not monic, such as 2x - 3, students often panic. They think the method fails. It does not. You simply divide the leading term of the dividend by the leading term of the divisor, exactly as before. Subtract, bring down, repeat.

Some sources suggest factoring out the leading coefficient to make the divisor monic, then using synthetic division. The quotient may have fractional coefficients, which is fine. Exact rational coefficients are the correct output, not decimals. This is the same guidance found in OpenStax Precalculus 2e on rational functions, where polynomial division precedes asymptote analysis.

Dividing Polynomials: Common Mistakes and How to Avoid Them

When you subtract, you are adding the opposite of each term in the product. Write the product, change every sign, then add. Do not try to subtract in your head. You must continue until the degree of the remainder is less than the degree of the divisor. Know which tool fits which job.

Polynomial Long Division Examples: What to Practice

To be test-ready, practice these five types of problems. First, a simple monic divisor like x + 2. Second, a missing term in the dividend. Third, a non-monic divisor like 2x - 3. Fourth, a divisor of degree 2, like the second worked example above. Fifth, a division where the dividend has a lower degree than the divisor, in which case the quotient is 0 and the entire dividend is the remainder.

Work each type twice. The first time, go slowly and check every subtraction. On a test, you have about five minutes per problem. If you cannot finish in that time, you have not drilled enough. Speed comes from repetition, not from intelligence. OpenStax College Algebra 2e provides additional problem sets in section 3.6, and working through ten of them is enough to internalize the method.