Education
Polynomial Long Division Calculator
Enter coefficients of the dividend and the divisor in descending order. The calculator runs polynomial long division and returns the quotient, the remainder, and the per-step working in a clear table.
Coefficients
Enter coefficients in descending order, including zeros for missing powers. For example, x^3 - 7x + 6 is 1, 0, -7, 6.
e.g. 1, 0, -7, 6
e.g. 1, -2 (for x - 2)
Step trace
| Step | Quotient term | Subtract (term x divisor) |
|---|---|---|
| 1 | 1x^2 | 1, -2 |
| 2 | 2x | 2, -4 |
| 3 | -3 | -3, 6 |
Each step picks a quotient term so the leading coefficient of the working remainder cancels, then subtracts the product. The process stops once the working polynomial has degree less than the divisor.
Quotient
x^2 + 2x - 3
Remainder: 0
Dividend = Divisor x Quotient + Remainder, with degree of Remainder less than degree of Divisor. The full identity holds term by term for any pair of polynomials.
Examples
(x^3 - 7x + 6) / (x - 2)
Quotient x^2 + 2x - 3, Remainder 0
(2x^3 + 3x^2 - x + 1) / (x^2 - 1)
Quotient 2x + 3, Remainder x + 4
(x^4 + 1) / (x^2 + 1)
Quotient x^2 - 1, Remainder 2
(x^2 + 3x + 2) / (x^3 + 1)
Quotient 0, Remainder x^2 + 3x + 2
How it works
At each step the algorithm picks the quotient term that cancels the leading coefficient of the working dividend, multiplies it by the divisor, and subtracts. The new working dividend has a lower degree, and the process repeats.
Identity · dividend = divisor x quotient + remainder
Step rule · qᵢ = lead(working) / lead(divisor)
Stops when the working polynomial's degree is less than the divisor's.
Related algebra calculators
- Synthetic division calculator for the compact shortcut when the divisor is exactly x - c.
- Factoring calculator for quadratic and difference-of-squares factoring over the integers.
- Long division calculator for integer long division (the same algorithm applied to numbers).
- Remainder calculator for the integer remainder counterpart and modulo arithmetic.
- All education calculators.
Frequently asked questions
Polynomial long division is the standard algorithm for dividing one polynomial by another. It produces a quotient polynomial and a remainder, just like integer long division. The identity is dividend = divisor x quotient + remainder, with the remainder of strictly lower degree than the divisor.
List the coefficients of each polynomial in descending order of power, separated by commas. Include zeros for any missing terms. The polynomial 2x^3 - 5x + 1 has coefficients 2, 0, -5, 1.
Use synthetic division when the divisor is exactly x - c (linear with leading coefficient 1) for the fastest path. Use polynomial long division for any other divisor, including 2x - 5, x^2 + 1, x^3 + x - 1, and so on.
Then division stops immediately: the quotient is 0 and the remainder is the dividend itself. The calculator reports both.
That is fine. Polynomial long division works over the rationals (and over the reals); coefficients may be any real number. The calculator shows up to 6 decimal places and integers when the value is exact.
If the remainder had degree at least equal to the divisor, you could keep dividing one more step. The algorithm stops exactly when no further reduction is possible, which is the definition of polynomial remainder.
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