How to Do Long Division of Polynomials- Complete Tutorial
What Long Division of Polynomials Actually Is
Long division of polynomials is the process of dividing a polynomial by another polynomial of lower degree. It's the same idea as regular long division, but with variables and exponents thrown in.
You need this when you want to simplify expressions, find remainders, or factor polynomials. Polynomial long division shows up in calculus, algebra, and engineering problems. If you're avoiding it, you're limiting yourself.
The Basic Setup
Before you start dividing, arrange everything in descending order by exponent. Every term must be present—even the ones with coefficient zero. Missing terms create gaps that mess up your work.
The dividend goes inside the division bracket. The divisor goes outside. You're finding how many times the divisor fits into the current terms of the dividend.
Step-by-Step Process
Step 1: Set Up the Problem
Write the dividend under the division symbol. Write the divisor to the left. Both polynomials must be in descending order with no missing degrees.
Example: Divide 2x³ + 7x² + 9 by x + 3
Notice the dividend has no x term. You need to write it as 2x³ + 7x² + 0x + 9.
Step 2: Divide the First Terms
Take the first term of the dividend (2x³) and divide it by the first term of the divisor (x). 2x³ ÷ x = 2x².
Write 2x² above the division bar, aligned with the x² term.
Step 3: Multiply and Subtract
Multiply the entire divisor by the term you just found (2x²). So (x + 3) × 2x² = 2x³ + 6x².
Subtract this from the dividend: (2x³ + 7x²) - (2x³ + 6x²) = x².
Bring down the next term (0x) to get x² + 0x.
Step 4: Repeat
Divide x² by x → x. Multiply the divisor by x → x² + 3x. Subtract → (x² + 0x) - (x² + 3x) = -3x.
Bring down the next term (9) → -3x + 9.
Step 5: Continue Until Done
Divide -3x by x → -3. Multiply the divisor by -3 → -3x - 9. Subtract → ( -3x + 9) - (-3x - 9) = 18.
The remainder is 18. Your answer is 2x² + x - 3 with remainder 18, or written as a mixed expression: 2x² + x - 3 + 18/(x+3).
Complete Worked Example
Let's divide x² + 5x + 6 by x + 2.
- Setup: Dividend is x² + 5x + 6. Divisor is x + 2.
- First division: x² ÷ x = x. Write x above the bar.
- Multiply: x(x + 2) = x² + 2x.
- Subtract: (x² + 5x) - (x² + 2x) = 3x. Bring down +6.
- Second division: 3x ÷ x = 3. Write 3 next to x.
- Multiply: 3(x + 2) = 3x + 6.
- Subtract: (3x + 6) - (3x + 6) = 0.
Answer: x + 3. No remainder. x² + 5x + 6 factors as (x + 2)(x + 3).
Comparing Division Methods
| Method | Best For | Requirements | Speed |
|---|---|---|---|
| Long Division | All polynomial divisions | Divisor with x term | Slower |
| Synthetic Division | Divisors like x - c or x + c | Divisor must be linear with coefficient 1 | Faster |
| Factoring | Simple trinomials | Polynomials that factor nicely | Fastest when it works |
Synthetic Division: The Faster Alternative
When your divisor is in the form x - c (or x + c), synthetic division cuts your work in half. No variables in your calculations—just numbers.
Using the same example: divide x² + 5x + 6 by x + 2.
The divisor x + 2 means c = -2.
- Write the coefficients: 1, 5, 6
- Bring down the 1
- Multiply -2 × 1 = -2. Add to 5 → 3
- Multiply -2 × 3 = -6. Add to 6 → 0
- Results: 1, 3, 0
Answer: x + 3. Same result, less writing.
Synthetic division only works when the divisor's leading coefficient is 1. If you have 2x + 1, you need long division.
Common Mistakes That Ruin Your Answer
- Forgetting to include missing terms. Always account for every power, even if the coefficient is zero.
- Misaligning terms. Keep like terms in the same column during subtraction.
- Dropping the remainder. The remainder is part of the answer. Write it as a fraction or state it explicitly.
- Sign errors when subtracting. Subtraction means changing every sign in the expression you're subtracting.
- Using synthetic division when you shouldn't. If the divisor doesn't start with x (coefficient must be 1), synthetic won't work correctly.
When to Use Polynomial Long Division
Polynomial long division is necessary when:
- Simplifying rational expressions
- Finding asymptotes in rational functions
- Checking if one polynomial is a factor of another
- Solving partial fraction decomposition problems
- Working with improper fractions where the numerator has higher degree
Practice Tips
Start with simple problems where the divisor fits evenly. Work toward problems with remainders once you're comfortable with the process.
Check your work by multiplying the quotient by the divisor and adding the remainder. You should get the original dividend.
If synthetic division applies to your problem, use it. It takes less time and produces fewer opportunities for arithmetic errors.
The Bottom Line
Long division of polynomials follows the same logic as numerical long division. Divide, multiply, subtract, bring down, repeat. The variables and exponents are just notation—the process is mechanical.
Master the basics before moving to synthetic division. Understanding why the steps work makes it easier to catch mistakes when they happen.