How to Divide a Polynomial by a Binomial- Step-by-Step

What Is Polynomial Division?

Polynomial division is the process of dividing a polynomial by another polynomial of lower degree. When the divisor is a binomial (two terms), you have two main approaches: long division and synthetic division.

Most students encounter this in algebra courses. It's not optional material—polynomial division shows up in calculus, rational expressions, and finding asymptotes. You need to actually know how to do it.

The Two Methods: Long Division vs Synthetic Division

Long division works for any binomial divisor. Synthetic division only works when the divisor is in the form x - c (where c is a constant and the coefficient of x is exactly 1).

Here's a quick comparison:

How to Do Long Division with Polynomials

This is the standard algorithm. It works every time, even when synthetic division won't.

Step 1: Set Up the Problem

Write the dividend (the polynomial being divided) under the long division symbol. Arrange terms from highest to lowest degree. If any degree is missing, insert a 0 as a placeholder.

For example, dividing 2x³ + 3x² - 5 by x + 2:

The dividend is missing an x term, so write it as: 2x³ + 3x² + 0x - 5

Step 2: Divide the Leading Terms

Take the leading term of the dividend (2x³) and divide it by the leading term of the divisor (x). This gives you 2x². Write this above the division bar.

Step 3: Multiply and Subtract

Multiply the entire divisor (x + 2) by the term you just found (2x²). This gives 2x³ + 4x². Subtract this from the dividend:

(2x³ + 3x²) - (2x³ + 4x²) = -x²

Step 4: Bring Down and Repeat

Bring down the next term (-5x becomes part of the new expression: -x² - 5x). Divide -x² by x to get -x. Multiply and subtract again.

Continue until the degree of the remaining expression is less than the degree of the divisor.

Step 5: Identify Your Answer

The expression above the division bar is your quotient. Any remainder gets written over the original divisor as a fraction.

Final answer: 2x² - x - 2 + (1/(x + 2))

Synthetic Division: The Faster Method

When your divisor is x - c, synthetic division cuts the work in half. No variable writing, no aligning terms—just numbers.

How Synthetic Division Works

Using the same example (2x³ + 3x² - 5 divided by x + 2):

First, rewrite the divisor as x - (-2). The value c is -2.

Write the coefficients of the dividend: 2, 3, 0, -5 (include 0 for the missing x term).

Bring down the 2. Multiply by -2 → -4. Add to 3 → -1. Multiply by -2 → 2. Add to 0 → 2. Multiply by -2 → -4. Add to -5 → -4.

The bottom row (except the last number) gives you the quotient coefficients: 2, -1, 2. The last number is your remainder: -4.

Your answer: 2x² - x + 2 - (4/(x + 2))

Common Mistakes to Avoid

• Missing terms: Always check for gaps in degree. Insert 0 for every missing term or you'll get the wrong answer.
• Wrong sign in synthetic: Remember that for x + 2, you use -2 in the synthetic box. For x - 5, you use +5.
• Forgetting the remainder: If you have a remainder, it must appear in the final answer as a fraction.
• Dropping degrees: When writing your quotient, start with the correct degree. It should be one less than the dividend's degree.

Practice Problems to Try

Work through these to build speed:

• (x² - 4) ÷ (x - 2)
• (3x³ + 6x² - 9x) ÷ (3x)
• (2x⁴ - 5x² + 3) ÷ (x - 1)
• (x³ + 2x² - 17x + 10) ÷ (x + 5)

When You'll Actually Use This

Polynomial division isn't just busywork. You'll encounter it when:

• Simplifying rational expressions in precalculus
• Finding oblique asymptotes of rational functions
• Factoring polynomials (especially higher-degree ones)
• Verifying that x = a is a root of a polynomial
• Setting up partial fraction decomposition

The long division method is the foundation. Learn it properly first. Once you're comfortable, synthetic division becomes a shortcut worth using whenever the divisor fits the form x - c.