Polynomial Long Division Calculator
Divide one polynomial by another and follow every step in a clear long-division layout. Enter your expressions to find the exact quotient and remainder, see each multiplication and subtraction, and verify the answer.
Your Polynomial Division Result
Enter a dividend and a nonzero divisor, or choose an example to see the exact result and long-division steps.
- Quotient Q(x)
- Remainder R(x)
Final Answer
Check the Remainder
Polynomial Long Division Steps
Each column represents one power of x. Subtract the entire parenthesized product row, including its signs. Scroll within the notebook when needed.
Numbered explanation — Show calculation
Verification
How to Divide Polynomials Using Long Division
Polynomial long division uses the same basic process as long division with numbers. Start with the highest-degree terms, divide to find the next term of the quotient, multiply the entire divisor by that term, and subtract. Repeat the process with the new remainder until no further polynomial term can be divided.
Step 1: Arrange Both Polynomials
Write each polynomial in descending powers of x. If a power is missing, leave space for it or use a zero coefficient to keep the terms aligned. For example, x³ + 2x + 1 has a zero coefficient for x².
Step 2: Divide the Leading Terms
Divide the leading term of the current dividend by the leading term of the divisor. Write the result above the division bracket. This becomes the next term of the quotient.
Step 3: Multiply and Subtract
Multiply the complete divisor by the new quotient term. Place the product beneath the matching powers of x and subtract it from the current dividend. Be careful to distribute the subtraction sign to every term.
Step 4: Repeat Until the Remainder Is Smaller
Continue dividing, multiplying, and subtracting. Stop when the remainder is zero or its degree is lower than the degree of the divisor.
Step 5: Verify the Answer
Multiply the quotient by the divisor and add the remainder. The result must equal the original dividend exactly.
Polynomial Long Division Formula
For a polynomial P(x) and a nonzero divisor D(x), the division algorithm gives a unique quotient Q(x) and remainder R(x):
P(x) = D(x)Q(x) + R(x)
The remainder is either zero or has a degree smaller than D(x). When R(x) = 0, the divisor divides the dividend evenly. Otherwise, the result can also be written as Q(x) + R(x)/D(x), where D(x) ≠ 0.
Worked Polynomial Long Division Examples
These worked examples use fixed sample values. Enter your own expressions in the calculator above to generate a customized solution.
Example 1: Divide a Cubic Polynomial by a Linear Polynomial
Divide 2x³ − 3x² + 4x − 5 by x − 2.
The quotient is 2x² + x + 6, and the remainder is 7.
Check: (x − 2)(2x² + x + 6) + 7 = 2x³ − 3x² + 4x − 5.
The remainder is not zero, so x − 2 is not a factor of the dividend.
Example 2: Divide a Quartic Polynomial by a Quadratic
Divide x⁴ − 1 by x² + 1.
The quotient is x² − 1, and the remainder is 0.
Check: (x² + 1)(x² − 1) = x⁴ − 1.
This example shows that polynomial long division handles higher-degree divisors, not just linear expressions.
Example 3: Divide by a Non-Monic Linear Polynomial
Divide 6x³ + 11x² − 31x + 15 by 3x − 2.
The quotient is 2x² + 5x − 7, and the remainder is 1.
Check: (3x − 2)(2x² + 5x − 7) + 1 = 6x³ + 11x² − 31x + 15.
The leading coefficient of the divisor does not need to be 1.
Polynomial Long Division vs. Synthetic Division
Polynomial long division works with any nonzero polynomial divisor. Synthetic division is a shorter coefficient-based method used for linear divisors. If you are dividing by x − c, synthetic division may be faster. If the divisor is quadratic or higher-degree, use polynomial long division.
Common Polynomial Division Mistakes
- Missing powers: Use an empty place or a zero coefficient for any missing power of x. Misaligned powers produce incorrect subtraction.
- Incorrect subtraction signs: Subtract every term of the product, including negative terms.
- Stopping too early: Continue until the remainder is zero or has a degree lower than the divisor.
- Ignoring the remainder: A nonzero remainder is part of the answer. Include it in the division identity or the final rational expression.
Methodology and Limits
Both calculators use exact rational polynomial arithmetic rather than floating-point approximation. Polynomial long division repeatedly subtracts a multiple of the divisor until the remainder has a lower degree. Synthetic division uses a coefficient recurrence for linear divisors. Both results are checked using the identity P(x) = D(x)Q(x) + R(x). For a linear divisor ax + b, the remainder is also verified by evaluating P(−b/a).
The calculators accept univariate polynomials in x with supported rational coefficients and nonnegative integer exponents. They do not perform general symbolic algebra, multivariable division, or division by the zero polynomial. Degree is limited to 16, input to 1,200 characters and 400 tokens, nesting to 32 levels, numeric literals to 100 digits, and intermediate integers to 4,096 bits. Ambiguous forms such as 1/2x must be written as (1/2)x. Bounds also apply during expansion.
Reviewed: October 2026. Read our methodology and sources.
Frequently Asked Questions
What is polynomial long division?
Polynomial long division is a method for dividing one polynomial by another. It repeatedly divides leading terms, multiplies by the divisor, and subtracts until the remaining polynomial has a smaller degree than the divisor.
Can I divide a polynomial by a quadratic polynomial?
Yes. Polynomial long division works with any nonzero polynomial divisor, including quadratic and higher-degree expressions. Synthetic division is normally used for linear divisors instead.
What happens when a polynomial has missing terms?
Missing powers have zero coefficients. For example, x³ + 2x + 1 can be written as x³ + 0x² + 2x + 1. Keeping that zero position makes the long-division columns align correctly.
What if the divisor has a coefficient other than 1?
Long division still works. When dividing the leading terms, divide their coefficients as well as their powers of x. The result may contain fractions.
What if the dividend has a lower degree than the divisor?
The quotient is zero, and the remainder is the original dividend. This satisfies the division algorithm because the remainder already has a smaller degree than the divisor.
How do I check polynomial long division?
Multiply the divisor by the quotient and add the remainder. The result should exactly reproduce the original dividend: P(x) = D(x)Q(x) + R(x).
What does a zero remainder mean?
A zero remainder means that the divisor divides the dividend evenly. Therefore, the divisor is a polynomial factor of the dividend.
Is synthetic division the same as polynomial long division?
No. Both produce the same quotient and remainder when applied correctly, but their procedures differ. Synthetic division uses a compact coefficient table for a linear divisor, while long division handles polynomial divisors of any degree.
Can this calculator work with fractions?
Yes. Supported rational coefficients are calculated exactly, so results such as 1/2 or 3/4 are preserved as fractions rather than rounded decimal approximations.
Sources
OpenStax — Algebra and Trigonometry 2e: Dividing Polynomials; OpenStax — Precalculus 2e: Dividing Polynomials.