Polynomial Division Calculator
Divide one polynomial by another, inspect every long-division step, verify the identity, and export the current calculation as a validated Excel workbook.
Polynomial inputs
Real coefficients, decimal point notationLive result
Updates as coefficients changeDivision summary
Canonical model checksPolynomial long-division steps
Divide, multiply, subtract, repeat| Step | Current polynomial | Quotient term | Product subtracted | New remainder |
|---|---|---|---|---|
| 1 | x^4 - 27x^3 + 239x^2 - 753x + 540 | x^3 | x^4 - x^3 | -26x^3 + 239x^2 - 753x + 540 |
| 2 | -26x^3 + 239x^2 - 753x + 540 | -26x^2 | -26x^3 + 26x^2 | 213x^2 - 753x + 540 |
| 3 | 213x^2 - 753x + 540 | 213x | 213x^2 - 213x | -540x + 540 |
| 4 | -540x + 540 | -540 | -540x + 540 | 0 |
How to use this polynomial division calculator
What this calculator does. It divides a one-variable polynomial P(x) by a nonzero polynomial Q(x), producing a quotient A(x) and a remainder R(x) that satisfy the exact identity P(x) = Q(x) × A(x) + R(x). It also records the successive divide-multiply-subtract steps used by polynomial long division. The calculator works with real decimal coefficients through degree 10. It does not factor the quotient, solve for roots, or decide which algebraic method is best for a larger problem.
When to use it. Use it to check homework after performing long division by hand, simplify a rational expression before further analysis, test whether a proposed divisor is a factor, or inspect the remainder used by the Remainder and Factor Theorems. OpenStax provides a detailed explanation of polynomial long division and its quotient-remainder identity.
How to calculate
- The calculator opens with a complete demonstration: P(x) = x^4 - 27x^3 + 239x^2 - 753x + 540 and Q(x) = x - 1. Its results and validated Excel workbook are available immediately.
- Choose Dividend degree and Divisor degree. The coefficient fields update to include every power from the selected degree down to the constant term.
- Replace each coefficient. Use a period as the decimal separator, such as 2.5 or -0.75. Enter 0 for a missing term rather than omitting the field. Scientific notation and comma decimal notation are intentionally rejected.
- Read Quotient A(x), Remainder R(x), Result form, and the long-division table. The summary confirms the quotient degree, remainder degree, and identity check.
- Select Download Excel to export the current canonical inputs, outputs, and steps. Reset clears the demonstration coefficients; export then remains unavailable until every required coefficient is completed again.
Input guide
Dividend degree is a required whole number from 0 to 10 and identifies the highest power of x in P(x). For the startup example it is 4. Raising the degree adds coefficient fields; the new leading coefficient must be nonzero. A common mistake is selecting degree 4 while entering 0 for a4, which actually describes a lower-degree polynomial and is rejected.
Dividend coefficients a0 through a10 are required finite real numbers for the powers displayed. The accepted format is an optional sign, digits, and an optional period decimal; examples include 1, -27, 0, and 3.125. The coefficient matching the selected degree must have magnitude of at least 0.000000000001, while all lower coefficients may be zero. Changing any coefficient changes the current polynomial, the quotient terms, and possibly the remainder.
Divisor degree is a required whole number from 0 to 10 and identifies the highest power of x in Q(x). The example uses 1. A divisor may have higher degree than the dividend; in that boundary case the quotient is 0 and the entire dividend is the remainder. The selected divisor can never be the zero polynomial.
Divisor coefficients b0 through b10 follow the same numeric format. The coefficient matching the selected divisor degree is required with magnitude of at least 0.000000000001; lower coefficients may be zero. For example, b1 = 1 and b0 = -1 represent x - 1. Do not type the variable, exponent, plus sign between terms, or a full expression into a coefficient field.
Output guide
Quotient A(x) is the polynomial that multiplies Q(x) during division. Remainder R(x) is what remains after its degree becomes lower than the divisor's degree. Result form displays A(x) alone for exact division, or A(x) + R(x) / Q(x) when a nonzero remainder exists. Division says Exact only when every remainder coefficient is zero within the calculator's numerical tolerance.
Quotient degree is the highest nonzero power in A(x), or “Zero polynomial” when the quotient is 0. Remainder degree is similarly reported, with “No remainder” for exact division. Identity verified confirms that multiplying Q(x) by A(x) and adding R(x) reproduces P(x) coefficient by coefficient. The Steps pill and the five table columns show the current polynomial, quotient term, product subtracted, and new remainder at each iteration.
Worked example
With the startup values, divide x^4 - 27x^3 + 239x^2 - 753x + 540 by x - 1. The leading-term ratio x^4 ÷ x gives x^3. Subtracting x^3(x - 1) leaves -26x^3 + 239x^2 - 753x + 540. Repeating produces quotient terms -26x^2, 213x, and -540. The final remainder is 0, so the displayed quotient is exactly x^3 - 26x^2 + 213x - 540. Multiplying that quotient by x - 1 reconstructs the original dividend.
How polynomial long division works
At each iteration, the algorithm divides the leading term of the current polynomial by the leading term of the divisor. That ratio becomes the next quotient term. Multiplying the full divisor by that quotient term creates a polynomial whose leading term cancels when subtracted. Because the degree decreases after every successful step, the process terminates after finitely many iterations. The final remainder must have lower degree than Q(x). Khan Academy's polynomial division lesson collection offers additional guided practice.
Exact division, factors, and remainders
A zero remainder means Q(x) divides P(x) exactly, so Q(x) is a polynomial factor of P(x). For a linear divisor x - c, the remainder equals P(c), which connects long division to the Remainder Theorem and factor testing. OpenStax's lesson on the Remainder and Factor Theorems explains that relationship. A nonzero remainder is not an error; it is the unique lower-degree polynomial required by the quotient-remainder identity.