Divide polynomials by long division or synthetic division, with every step, the quotient and remainder in exact fractions, checked against OpenStax College Algebra 2e
How Long Division Works
- Divide the leading term of what is left by the divisor’s leading term: 6x³ ÷ 3x = 2x².
- Multiply the whole divisor by it: 2x²(3x − 2) = 6x³ − 4x².
- Subtract and bring down the next term: (6x³ + 11x²) − (6x³ − 4x²) = 15x², then 15x² − 31x.
- Repeat: 15x² ÷ 3x = 5x, leaving −21x + 15; −21x ÷ 3x = −7, leaving 1.
The remainder 1 has lower degree than 3x − 2, so the division stops: quotient 2x² + 5x − 7, remainder 1. Check: (3x − 2)(2x² + 5x − 7) + 1 = 6x³ + 11x² − 31x + 15.
How Synthetic Division Works
For a divisor x − c, write only the coefficients and c. OpenStax Example 3 divides 5x² − 3x − 36 by x − 3, so c = 3:
- Bring down the first coefficient: 5.
- Multiply by c and add to the next coefficient: 5 × 3 = 15; −3 + 15 = 12.
- Repeat: 12 × 3 = 36; −36 + 36 = 0.
- The bottom row 5, 12 | 0 gives the quotient 5x + 12 and remainder 0.
Write a 0 for any missing power: in Example 5, −9x⁴ + 10x³ + 7x² − 6 has no x term, so its coefficients are −9, 10, 7, 0, −6. For a divisor like 3x − 2, use c = 2/3 and divide the bottom row by 3; the calculator does this automatically.
OpenStax Examples, Reproduced
The first six rows are the section’s worked examples; the last two are its “Try It” exercises, whose answers are computed here.
The Remainder Theorem
Dividing by x − c leaves a remainder equal to the polynomial’s value at c. In Example 5 the remainder is 2, and −9 + 10 + 7 − 6 = 2. A remainder of 0 means x − c is a factor, which is how Example 6 finds the box’s height x² + x − 9 from its volume.
Common Mistakes
- Skipping missing terms. Without a placeholder 0, every later column shifts.
- Using the wrong sign for c. For x + 2, c is −2.
- Subtracting only the first term. Distribute the minus sign across the whole product.
- Using synthetic division for a quadratic divisor. It only works for linear divisors; use long division.
- Forgetting to divide by the leading coefficient when synthetic division is used with a divisor like 3x − 2.
What This Calculator Does Not Cover
It divides polynomials in one variable, x, with numeric coefficients (integers, decimals or fractions). Multivariable polynomials, complex coefficients and factoring a polynomial completely are outside its scope.
Frequently Asked Questions
How do you do polynomial long division?
Divide the leading terms, multiply the divisor by the result, subtract, bring down the next term, and repeat until the remainder has a lower degree than the divisor.
When can you use synthetic division?
When the divisor is linear, of the form x − c (or ax − b, dividing the result by a). For higher-degree divisors use long division.
What is (6x³ + 11x² − 31x + 15) ÷ (3x − 2)?
2x² + 5x − 7 with remainder 1, as in OpenStax College Algebra 2e Example 2.
How do I handle a missing term?
Write it with a 0 coefficient, e.g. −9x⁴ + 10x³ + 7x² + 0x − 6, so the columns line up.
What does a remainder of 0 mean?
The divisor is a factor of the dividend. For x − c, it also means c is a root.
How do I write the answer with a remainder?
Quotient + remainder ÷ divisor, e.g. 2x² + 5x − 7 + 1/(3x − 2).
How to Use This Calculator
- Enter the dividend — the polynomial being divided, e.g. 6x^3 + 11x^2 - 31x + 15.
- Enter the divisor — e.g. 3x - 2 or x^2 + 1.
- Choose long or synthetic division — synthetic works for linear divisors.
- Read the steps — each divide, multiply and subtract step (or the synthetic row), then the quotient and remainder.
📚 Sources & References- [1] OpenStax. College Algebra 2e, 5.4 Dividing Polynomials (2021; accessed October 2026). openstax.org
- [2] OpenStax. College Algebra 2e, 5.5 Zeros of Polynomial Functions: Remainder Theorem (2021; accessed October 2026). openstax.org
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