Divide polynomials with long division and synthetic division support for linear divisors. Get the quotient, remainder, and step-by-step algebra work.
For (x^3 – 6x^2 + 11x – 6) ÷ (x – 1), the quotient is x^2 – 5x + 6 and the remainder is 0.
Exact division
- Match leading terms.
- Multiply the divisor by that term.
- Subtract and repeat until the remainder degree is smaller than the divisor degree.
All calculations run locally in your browser. Your polynomial inputs are not sent to an external service.
How to Use the Polynomial Division Calculator
Enter the polynomial you want to divide in the dividend box and the polynomial you are dividing by in the divisor box. The calculator arranges terms by descending powers, divides leading terms, subtracts each product, and stops when the remainder has smaller degree than the divisor.
- Write powers with a caret, such as x^3.
- Use the same variable, x, in both polynomials.
- Check the quotient and remainder identity before using the result in another problem.
Polynomial Division Rule
Polynomial division follows the same structure as integer division. The dividend equals the divisor multiplied by the quotient plus the remainder.
The degree of the remainder must be lower than the degree of the divisor. If the remainder is 0, the divisor is a factor of the dividend.
When Synthetic Division Applies
Synthetic division is a shortcut for divisors that are linear, such as x – 2 or x + 5. The calculator still returns the long-division result, and it labels the method as synthetic-ready when the divisor is linear.
For x – c, synthetic division uses c as the root. For x + 2, the value is -2. This is also the number used in the remainder theorem.
Polynomial Division Examples
(x^3 – 6x^2 + 11x – 6) ÷ (x – 1)
The quotient is x^2 – 5x + 6 and the remainder is 0, so x – 1 is a factor.
(2x^3 + 3x^2 – 11x – 6) ÷ (x + 2)
The calculator divides leading terms, subtracts each product, and reports whether any constant remainder is left.
(x^4 – 1) ÷ (x^2 – 1)
This division produces x^2 + 1 with no remainder because x^4 – 1 factors as (x^2 – 1)(x^2 + 1).
Common Polynomial Division Mistakes
Leaving out zero-power terms
If a polynomial skips a power, treat that missing coefficient as 0. For example, x^3 – 2x + 5 has no x^2 term.
Subtracting the product incorrectly
After multiplying the divisor by the new quotient term, subtract the entire product from the current remainder.
Stopping too early
Continue until the remainder degree is smaller than the divisor degree. A constant remainder is valid when dividing by a linear polynomial.
Polynomial Division Calculator FAQs
Can this divide by a quadratic?
Yes. The calculator supports polynomial divisors such as x^2 – 1 as long as the expression is written in x terms.
Does it show synthetic division?
For linear divisors, it identifies the synthetic root and shows a synthetic-ready note. The result is still verified by polynomial long division.
What does a zero remainder mean?
A zero remainder means the divisor divides the dividend exactly, so the divisor is a factor of the dividend.
Can I enter fractions?
This version supports integer and decimal coefficients. Fraction-style coefficient input can be added later if needed.
Related SolverCalc tools
Reviewer
This educational math content is reviewed under the SolverCalc reviewer rule by Prof. Dr. Khalil Mudassar. WordPress adds the author bio automatically, so this page does not include a duplicate author box.
Keep solving with SolverCalc
Use related algebra tools to factor, simplify, or check the result after polynomial division.
Browse Algebra ToolsSources
View mathematical references
Disclaimer
This calculator is intended for general educational and practical use. Review important algebra results before relying on them in graded or professional work.