Polynomial Division Calculator
Enter a dividend and divisor polynomial in terms of x to compute the quotient and remainder using polynomial long division.
What is polynomial division?
Polynomial division is the process of dividing one polynomial by another to find two outputs: a quotient polynomial and a remainder polynomial. It works just like arithmetic long division, except the “digits” are now terms such as x², x³, and constants.
The central identity is:
Dividend = Divisor × Quotient + Remainder
where the remainder always has degree strictly lower than the degree of the divisor.
How to use this polynomial calculator division tool
- Type the first polynomial in the Dividend box.
- Type the second polynomial in the Divisor box.
- Click Calculate Division to compute quotient and remainder instantly.
- The identity check confirms your result in expanded form.
Supported input examples
x^4 - 5x^2 + 4
3x^3 + x - 9
-2x^5 + 7x^2 - x + 1
x - 2
Polynomial long division in 4 steps
1) Divide leading terms
Take the highest-degree term in the current dividend and divide by the highest-degree term in the divisor. That gives the next term of the quotient.
2) Multiply back
Multiply the new quotient term by the full divisor polynomial.
3) Subtract
Subtract the multiplied expression from the current dividend row.
4) Repeat
Continue until the degree of what remains is lower than the degree of the divisor. The leftover expression is the remainder.
Worked example
Suppose we divide x^3 - 6x^2 + 11x - 6 by x - 1.
- Leading term division: x^3 / x = x^2
- Then continue with subtraction steps
- Final quotient: x^2 - 5x + 6
- Remainder: 0
Since remainder is zero, x - 1 is an exact factor.
When to use polynomial division
- Factoring higher-degree expressions
- Simplifying rational expressions
- Finding asymptotes in algebra/calculus
- Using the Remainder Theorem and Factor Theorem
- Preparing for synthetic division shortcuts
Common mistakes to avoid
- Forgetting missing terms (for example, skipping an x² term without treating it as coefficient 0).
- Sign errors during subtraction.
- Using a divisor of zero.
- Mixing variables (this tool uses only x).
Final tip
Always verify with the identity Dividend = Divisor × Quotient + Remainder. This calculator performs that check for you so you can trust the result while still learning the process.