Dividing Polynomials
8 min read · Updated September 26, 2026 · by the SolveCalcPro editorial team
Dividing polynomials works like long division of numbers, term by term. It helps you factor higher-degree polynomials, simplify algebraic fractions and understand remainders. There are two main methods: polynomial long division, which always works, and synthetic division, a shortcut for linear divisors.
Dividing by a monomial
Divide each term separately: (6x³ + 9x² − 3x) ÷ 3x = 2x² + 3x − 1.
Polynomial long division
Divide (x² + 5x + 6) by (x + 2):
- Divide the first terms: x² ÷ x = x. Write x above.
- Multiply x by (x + 2): x² + 2x. Subtract: (x² + 5x) − (x² + 2x) = 3x. Bring down +6 to get 3x + 6.
- Divide the first terms again: 3x ÷ x = 3. Write +3.
- Multiply 3 by (x + 2): 3x + 6. Subtract: 0.
The quotient is x + 3 with no remainder. Another: (x³ − 1) ÷ (x − 1) = x² + x + 1.
Handling remainders
If the last subtraction is not zero, the leftover is the remainder. The full answer is written quotient + remainder ÷ divisor. For (2x³ + 3x² − 5x + 4) ÷ (x − 1), the quotient is 2x² + 5x and the remainder is 4, so it equals 2x² + 5x + 4/(x − 1).
Synthetic division for (x − c)
When the divisor is (x − c), list the coefficients and use c. For the example above, with c = 1: 2 | 3 | −5 | 4 gives bottom row 2, 5, 0, 4. The first three numbers are the quotient’s coefficients (2x² + 5x + 0) and the last is the remainder. Read more in the remainder theorem guide.
Check your work
Multiply the quotient by the divisor and add the remainder; you should get the original polynomial. (x + 3)(x + 2) = x² + 5x + 6 ✓. The polynomial calculator can do the multiplication for you.
Practice questions
Test yourself, then tap to check. Answers are calculated by the tools linked below.
- Multiply (x + 3)(x − 5).
Show answer
Product: x² − 2x − 15
- Solve 4x − 3 = x + 12.
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x: 5
- Find the roots of f(x) = x² − 6x + 8.
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Function: f(x) = x^2 - 6x + 8
Frequently asked questions
When can I use synthetic division?
When dividing by a linear factor of the form (x − c).
What if a term is missing in the dividend?
Insert it with a zero coefficient, such as 0x², so the columns line up.
How do I know the division is exact?
The remainder is zero, which means the divisor is a factor.
What is the degree of the quotient?
The degree of the dividend minus the degree of the divisor.
Try it yourself
Polynomial CalculatorAdd, subtract and multiply polynomials and evaluate the result at any x.Equation SolverSolve linear equations and 2×2 systems of equations with steps and a built-in check.Graphing CalculatorPlot any function of x, find its roots and y-intercept and see its highest and lowest values.Keep learning
The Remainder and Factor Theorems7 min readFactoring Polynomials9 min readAlgebraic Fractions8 min readWritten and reviewed by the SolveCalcPro editorial team. Spot a mistake? Tell us. See our editorial policy.