Polynomial Long Division
Polynomial long division splits a dividend into a quotient plus a remainder over the divisor, just like dividing whole numbers. Enter both polynomials in x using ^ for exponents; the calculator returns the quotient and remainder.
Splitting a fraction into a polynomial and a remainder
Dividing 17 by 5 gives 3 with 2 left over, and no amount of further work reduces the remainder below the divisor. Polynomial division works the same way, with degree taking the place of size: divide until what remains has lower degree than the divisor, and stop.
This calculator performs that division, reporting the quotient and the remainder separately along with the steps that produced them.
The result is best read as a rewriting rather than a calculation. A rational expression whose numerator has the higher degree cannot be handled by most techniques — partial fractions refuses it, and its behaviour at large values is not obvious. Division turns it into a polynomial plus a proper fraction, and both pieces are then tractable.
The remainder also answers a question of its own. A remainder of zero means the divisor divides the dividend exactly, which is a factor test — and for a linear divisor it is the remainder theorem in action.
How to use this calculator
- Enter the dividend The polynomial being divided, in x with ^ for exponents. Missing powers need not be written as zero terms.
- Enter the divisor The polynomial dividing into it. It must not be the zero polynomial, which is rejected explicitly.
- Follow the division steps Each line records one term brought into the quotient and the subtraction that followed it. The number of steps is the difference in degree plus one.
- Read the quotient and remainder Reported on separate lines. The remainder always has lower degree than the divisor, or is zero.
The formula, and where it comes from
dividend = divisor × quotient + remainder, with deg(remainder) < deg(divisor)
That identity is the whole content of the operation, and it is the check worth running on any answer: multiply the divisor by the quotient, add the remainder, and the dividend must come back exactly.
The degree condition is what makes the answer unique. Without it, any number of pairs would satisfy the identity; requiring the remainder to be smaller in degree than the divisor pins down exactly one quotient and one remainder.
The algorithm mirrors long division with numbers. At each step the leading term of what remains is divided by the leading term of the divisor, that quotient term is recorded, and the divisor multiplied by it is subtracted — which cancels the leading term and lowers the degree by one.
The loop continues while the working polynomial still has degree at least the divisor's. Once it drops below, nothing further can be extracted and what is left is the remainder. Leading coefficients that fall within rounding distance of zero are trimmed before the result is reported.
What each input means
- dividend Dividend — form field “Dividend”
- The polynomial being divided. Its degree relative to the divisor's determines how many steps the division takes.
- divisor Divisor — form field “Divisor”
- The polynomial dividing into it. Must be non-zero; its degree sets the ceiling on the remainder's.
- quotient Quotient
- The polynomial part of the answer. Its degree is the difference between the two input degrees.
- remainder Remainder
- What is left over, always of lower degree than the divisor. Zero exactly when the divisor is a factor.
Worked examples
Every number below is produced by the same calculation engine the tool above runs. Nothing here is typed by hand, so the walkthrough cannot drift from what you get when you enter the same values yourself.
A cubic divided by a linear factor
A cubic with a missing linear term divided by a linear polynomial. The division does not come out exactly.
Inputs Dividend = x^3 - 2x^2 - 4, Divisor = x - 3
- Dividend x³ − 2x² − 4
- Divisor x − 3
- Divide leading terms bring down 1·x², then subtract
- Divide leading terms bring down 1·x¹, then subtract
- Divide leading terms bring down 3·x⁰, then subtract
- Quotient x² + x + 3
- Remainder 5
Result quotient = x² + x + 3, remainder = 5
The dividend has no x term, and the algorithm handles that without it being written in — the coefficient list simply carries a zero there. Writing out missing powers is a hand-working habit, not a requirement of the method.
The remainder is a non-zero constant, so the divisor is not a factor. By the remainder theorem that constant is also the value of the dividend at x = 3, which is a much quicker way to obtain it than dividing.
An exact division
A difference of squares divided by one of its factors. The remainder should be zero.
Inputs Dividend = x^2 - 1, Divisor = x - 1
- Dividend x² − 1
- Divisor x − 1
- Divide leading terms bring down 1·x¹, then subtract
- Divide leading terms bring down 1·x⁰, then subtract
- Quotient x + 1
- Remainder 0
Result quotient = x + 1, remainder = 0
The remainder is zero, which confirms the divisor is a factor. The quotient is the other factor, so this division has effectively factored the dividend.
That is the general pattern: once one factor of a polynomial is known, dividing by it produces a lower-degree polynomial whose own factors complete the job. It is how a cubic gets factored after one root is spotted.
Reading the result
A zero remainder is a factor test
It means the divisor divides the dividend exactly, so the divisor and the quotient are both factors. That is the standard route from a known root to a complete factorisation.
Degrees always add up
The quotient's degree is the dividend's minus the divisor's. A quotient of unexpected degree means a term was lost, and checking this before reading the remainder catches most errors.
The remainder theorem shortcut
When the divisor is x minus a number, the remainder equals the dividend evaluated at that number. One substitution replaces the whole division whenever the quotient itself is not needed.
When you would use this
Preparing an improper rational expression
Partial-fraction decomposition requires the numerator's degree to be lower than the denominator's. Division extracts the polynomial part first, leaving a proper fraction that can then be split.
Factoring a higher-degree polynomial
Once one root is found, dividing by the corresponding linear factor reduces the problem by a degree. Repeating that eventually brings a cubic or quartic down to a quadratic.
Assumptions and limitations
What this calculator assumes
- Both inputs are polynomials in x with non-negative whole exponents.
- The divisor is not the zero polynomial.
- Division stops when the working polynomial's degree falls below the divisor's.
- Missing powers are treated as zero coefficients without needing to be written.
Where it stops being the right tool
- One variable only: multivariate polynomial division is out of scope.
- Coefficients are shown as decimals, so a quotient with fractional coefficients appears rounded rather than exact.
- The full long-division layout is not drawn — the steps are described rather than tabulated.
- Synthetic division is not offered separately, though it gives the same result for a linear divisor.
Common mistakes
Forgetting a missing power when working by hand
Why it happens. A dividend written without its x term looks complete, and the columns of a hand-worked division then shift by one place, corrupting everything after that point.
How to avoid it. Write missing powers as zero terms before starting. The calculator does this internally, which is why its answer can differ from a hand attempt that skipped them.
Adding instead of subtracting at each step
Why it happens. The subtraction is of a product that often contains negative coefficients, so two sign changes are in play at once and one is easily dropped.
How to avoid it. Multiply the divisor by the quotient term first, write the result down, then subtract deliberately. Verify at the end by reconstructing the dividend.
Continuing past the stopping point
Why it happens. The remainder still contains an x in most cases, which looks like something left to divide.
How to avoid it. Compare degrees. Once what remains is lower in degree than the divisor, the division is finished by definition.
Frequently asked questions
How do I enter the polynomials?
Use ^ for exponents and join terms with plus and minus, for example x^3 - 2x^2 - 4 divided by x - 3. Missing powers can be left out entirely — they are treated as zero coefficients internally.
What does the remainder tell me?
A remainder of zero means the divisor is a factor of the dividend, so the quotient is the other factor. A non-zero remainder is what is genuinely left over, and it always has lower degree than the divisor.
Is this the same as synthetic division?
It gives the same quotient and remainder. Synthetic division is a compressed shortcut that works only when the divisor is linear; long division handles a divisor of any degree.
How do I check the answer?
Multiply the divisor by the quotient and add the remainder. The dividend must come back exactly — that identity is what the division is defined to produce.