How to divide polynomials
What this calculator does
Use polynomial long division to split a dividend into a divisor times a quotient, plus a remainder. This calculator is for students checking algebra homework and teachers preparing worked examples. It accepts two expanded polynomials in x and shows the full arithmetic, including fractional coefficients. You do not need to supply zero terms or put terms in order. It combines like terms and arranges powers before dividing. The result is an exact polynomial identity, not a numerical value of x or a solution of an equation.
How to use the two input fields
Enter the polynomial being divided in Dividend and the polynomial you are dividing by in Divisor. For example, enter x^3 - 2x^2 - 4x + 8 and x - 3. Press Calculate or press Enter in either input. Read the quotient and remainder first, then follow the steps below. Editing either field removes the old result so it cannot be mistaken for the new answer. The example buttons ask before replacing your inputs. Clear inputs removes both expressions and their result.
Input notation and exact coefficients
Write powers with a caret: x^3 means x cubed. Both 3x and 3*x are accepted. Use + and - between terms, and put a minus sign before a negative first term. Use a slash for a numerical fraction, such as 1/2x, or a decimal point, such as 0.5x. These two coefficients represent exactly the same rational number. Use decimal points in every interface language; a comma is not a decimal separator here. Parentheses, products of polynomials, functions, other variables, fractional or negative exponents, and scientific notation are outside the grammar. Expand a product yourself before entering it.
Why each long division step works
The highest-power term of the current remainder is divided by the highest-power term of the divisor. This produces the next term of the quotient. Multiply that term by the entire divisor, then subtract the entire product from the current remainder. The leading terms cancel, reducing the degree. Repeat until the remainder is zero or has a smaller degree than the divisor. Each card shows these three operations. Missing powers behave as zero coefficients, even though the display omits them. Every subtraction uses the current remainder, not the original dividend.
Example 1: a nonzero remainder
For (x^3 - 2x^2 - 4x + 8) divided by (x - 3), the quotient is x^2 + x - 1 and the remainder is 5. The successive remainders are x^2 - 4x + 8, then -x + 8, then 5. Check by multiplying (x - 3)(x^2 + x - 1) and adding 5: the original cubic returns. When writing the answer as a rational expression, use x^2 + x - 1 + 5/(x - 3), with x ≠ 3.
Example 2: fractional coefficients
Divide 1/2x^2 + 3/4x + 1/4 by x + 1. The quotient is 1/2x + 1/4 and the remainder is 0. The first quotient term is 1/2x; subtracting its product leaves 1/4x + 1/4. The next term, 1/4, cancels what remains. You may enter 0.5x^2 + 0.75x + 0.25 instead and obtain the same reduced fractions. This makes the tool useful for comparing a decimal exercise with a fraction-based worked answer without introducing rounding.
Limits and common mistakes
Each field allows 500 characters and 60 terms, with whole-number exponents from 0 to 20. Each numerical part allows nine digits, including digits after the decimal point; fraction numerator and denominator are counted separately. Exact intermediate numerators and denominators are limited to 2,000 digits. A divisor that simplifies to zero, such as x - x, is rejected. A fraction denominator cannot be zero either. If an input is rejected, keep the expression and correct the indicated field. Check signs carefully: subtracting a negative coefficient adds its positive value.
Copying and using the solution
Copy solution copies the simplified inputs, quotient, remainder, verified identity and all displayed steps as plain text. Powers use ^ and coefficients use reduced fractions, so the text stays readable in notes or messages. The expandable Solution text area provides manual copying if clipboard access is unavailable. Calculation happens in this browser without submitting your expressions to a solver service. This page does not save your work between visits. The common language menu warns before navigating because changing language resets the calculation.
Frequently asked questions
Why is my quotient zero?
If the dividend has a lower degree than the divisor, no leading term can be divided to produce a polynomial term. The quotient is zero and the entire dividend is the remainder. A zero dividend also gives a zero quotient and remainder for any nonzero divisor.
Can the divisor be a constant?
Yes. Any nonzero rational constant is supported. Every coefficient is divided by that constant and the remainder is zero. A constant divisor has degree zero.
Does a remainder of zero mean the divisor is a factor?
Yes, the identity then reads dividend = divisor × quotient. For a linear divisor x - c, the remainder also equals the dividend evaluated at c. This is the remainder theorem and offers a separate way to check that special case.
Is this synthetic division?
No. This tool uses long division for every supported divisor, including quadratic and higher-degree polynomials. It does not display a synthetic-division table or find roots.
What does “verified identity” establish?
The calculator checks all coefficients of dividend = divisor × quotient + remainder using exact fractions. The identity holds for every x. The expression dividend/divisor is only defined where the divisor is nonzero; cancelling a factor does not remove that original restriction.