Polynomial Division Calculator

Background.

Dividing a polynomial by a factor such as (x − 2) is the step that unlocks most factoring problems in algebra. If the division comes out exactly, you have found a root and reduced the polynomial by one degree; if it leaves something over, that leftover tells you the value of the polynomial at x = 2. Synthetic division is the shorthand for doing this by hand: instead of writing out the full long-division layout with every power of x, you work only with the coefficients, multiplying and adding in a single row.

The method rests on two results from algebra. The remainder theorem says that dividing P(x) by (x − r) leaves a remainder of exactly P(r). The factor theorem follows from it: (x − r) divides P(x) with nothing left over precisely when P(r) = 0, that is, when r is a root. So a remainder of zero is the signal you were looking for.

In practice that makes synthetic division a search tool. To factor x³ − 7x + 6, you try likely roots, here the divisors of 6, until one gives remainder 0. Trying r = 2 does, leaving the quadratic x² + 2x − 3, which factors at sight into (x + 3)(x − 1).

The page handles one variable and a divisor of the form (x − r) only. Division by a quadratic, or any divisor of degree two or more, needs full polynomial long division.

What is polynomial division calculator?

Synthetic division is a compact way of dividing a polynomial P(x) by a linear divisor (x − r). Working from the highest power down, each coefficient of the quotient is the next coefficient of P plus r times the previous result. The quotient has degree one less than P, and the final number produced is the remainder, which by the remainder theorem equals P(r). The procedure is the same arithmetic as Horner's method for evaluating a polynomial.

How to use this calculator.

1. 01Enter the dividend's coefficients from the highest power down, separated by commas or spaces: x² − 3x + 2 is 1, −3, 2.
2. 02Put a 0 in for every missing power. x³ − 7x + 6 has no x² term, so it is entered as 1, 0, −7, 6; leaving the zero out silently divides a different polynomial.
3. 03Enter r, the number subtracted from x in the divisor. For (x − 4) enter 4; for (x + 3) enter −3.
4. 04Read the remainder and the quotient coefficients, which are listed highest power first and start one degree below the dividend.

The formula.

For P(x) = aₙxⁿ + … + a₁x + a₀ divided by (x − r), set b₁ = aₙ, then repeat bₖ₊₁ = (next coefficient) + r × bₖ down the list. The b values before the last are the quotient's coefficients, from xⁿ⁻¹ downward; the final value is the remainder R. Together they satisfy P(x) = (x − r)·Q(x) + R. The remainder is also P(r), because substituting x = r makes the (x − r)·Q(x) term vanish. The quotient coefficients are shown unrounded; the remainder is rounded to ten decimal places. The divisor must have leading coefficient 1, so a divisor like (2x − 1) needs a small adjustment, explained below.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, Rice University. College Algebra 2e, 2021. Algebra and function chapters. Retrieved 2026-08-06.
2. [2]OpenStax, Rice University. Precalculus 2e, 2021. Trigonometry and analytic geometry chapters. Retrieved 2026-08-06.
3. [3]NIST/SEMATECH. e-Handbook of Statistical Methods, 2012. Chapter 1.3 and Chapter 4. Retrieved 2026-08-06.