How this calculation works
Divide leading terms to obtain a quotient term, subtract its product with the divisor, then repeat until the remainder is zero or has lower degree. Exact rational arithmetic prevents approximate cancellation from creating false residual terms.
Dividend = divisor × quotient + remainder; deg(remainder) < deg(divisor) when nonzeroA worked example
Dividing x² − 3x + 2 by x − 1 gives quotient x − 2 and remainder 0. The identity is (x − 1)(x − 2) = x² − 3x + 2.
Good to know
Bounded exact rational inputs within ±1,000,000, up to 6 decimal places or integer fraction denominators up to 1,000,000. Large intermediate fractions have a size limit. No arbitrary expression evaluation, multiple variables or general computer-algebra system. Approximate lengths and quadratic square roots are labeled separately.
Reference: OpenStax: polynomial division (opens in a new tab) ↗
The quotient alone does not describe a division with a remainder
The polynomial division identity is A = BQ + R, where B is nonzero and R is zero or has lower degree than B. It is an identity between polynomials, not a claim that R disappears.
Dividing x² − 3x + 2 by x−1 gives Q = x−2 and R = 0. Dividing x²+1 by x−1 instead gives Q = x+1 and R = 2. In the second case, the rational expression is x+1 + 2/(x−1) wherever the original denominator is nonzero.
At each step, dividing leading terms finds the next quotient term. Subtracting that term times the divisor cancels the current leading power. The calculator shows the successive remaining polynomials with exact coefficients.
Algebraic simplification does not restore a missing denominator input
Even when the remainder is zero, a rational expression can have an excluded input. For example, (x²−1)/(x−1) agrees with x+1 only where x ≠ 1. The polynomial identity x²−1 = (x−1)(x+1) remains valid at every x, but the original fraction does not.
A rational polynomial zero finder uses division after confirming that a candidate makes the remainder zero. Repeated division by the same linear factor identifies multiplicity, while any residual factor must still be retained.
A dividend of lower degree than the divisor has quotient zero and remainder equal to the dividend. A nonzero constant divisor simply scales every coefficient. A zero divisor is always rejected.
A few questions, answered
Is synthetic division being used?
The tool uses general exact polynomial long division, which also handles divisors beyond a monic linear factor.
Can there be a remainder with the divisor’s degree?
Not in the completed division algorithm. A nonzero remainder has strictly lower degree.
Does a zero remainder allow every x in the original fraction?
No. Original denominator zeros remain excluded when interpreting a rational function.