Polynomial Division Calculator — Long Division Step by Step
Enter the coefficients of the dividend and divisor polynomials (highest degree first) to perform polynomial long division and find the quotient and remainder with full step-by-step working.
Quotient: x - 5
Divide leading term: 1/1 = 1
Divide leading term: -5/1 = -5
Quotient
Remainder
- 1
Degree of dividend
2 - 2
Degree of divisor
1 - 3
Quotient degree
2 − 1 = 1
How does this calculator work?
Polynomial long division: at each step, divide the leading term of the remainder by the leading term of the divisor to get the next quotient term; multiply back and subtract. Repeat until the remainder degree is less than the divisor degree. The result: Dividend = Divisor × Quotient + Remainder. Enter coefficients in descending order, using 0 for missing powers.
Formula
How this is calculated
Polynomial long division works exactly like integer long division but with polynomials instead of digits. You enter the coefficients of both polynomials in descending degree order — for example, x² − 3x − 10 is entered as "1 -3 -10" and x + 2 as "1 2".
At each step, the leading term of the current remainder is divided by the leading term of the divisor to find the next term of the quotient. That term is multiplied by the entire divisor and subtracted from the remainder, reducing its degree by one. The process repeats until the degree of the remainder is less than the degree of the divisor, at which point the remainder is the final remainder.
The result satisfies Dividend = Divisor × Quotient + Remainder, just as for integers. If the remainder is zero, the divisor is a factor of the dividend. The calculator does exact arithmetic on the rational coefficients, so any accumulated rounding error for large polynomials is minimal. Synthetic division is a shortcut applicable only when the divisor is a linear binomial (x − c); this calculator handles divisors of any degree.
Frequently asked questions
Use a zero coefficient for the missing power. For example, x³ + x − 2 (no x² term) is entered as "1 0 1 -2". The zero placeholder keeps the degree alignment correct during division.
If the remainder is 0, the divisor is an exact factor of the dividend (a root of the divisor is also a root of the dividend). A non-zero remainder means the divisor does not divide evenly. The remainder theorem states that dividing a polynomial P(x) by (x − c) gives a remainder equal to P(c).
Synthetic division is a streamlined shorthand that works only when the divisor is a monic linear polynomial (x − c). It is faster by hand in that special case. This calculator uses full long division, which handles divisors of any degree and any leading coefficient.
Also known as
TG we-Calculate Editorial Team. (2026). Polynomial Division Calculator — Long Division Step by Step [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/polynomial-division-calculator
TG we-Calculate Editorial Team. "Polynomial Division Calculator — Long Division Step by Step." TG we-Calculate. 2026. https://we-calculate.com/calculator/polynomial-division-calculator.
TG we-Calculate Editorial Team, "Polynomial Division Calculator — Long Division Step by Step," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/polynomial-division-calculator
@misc{wecalculate_polynomial_division_calculator, title = {Polynomial Division Calculator — Long Division Step by Step}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/polynomial-division-calculator}}, year = {2026}, note = {TG we-Calculate} }
Did this calculator help you?