Polynomial division gives P = D × Q + R. The nonzero remainder has lower degree than the divisor. If R = 0, the divisor is a factor of the dividend. If the divisor has higher degree than the dividend, the quotient is 0 and the dividend itself is the remainder. A nonzero constant divisor always gives remainder 0.

Repeat: divide the leading terms, multiply the whole divisor, then subtract like powers. Stop when no further polynomial quotient term is possible. The exact identity is valid for every x; the expression P/D is defined only where D is not zero.

For division of numbers, use the long division calculator. For arithmetic with fractions, use the fraction calculator.

Use one variable, lowercase x, and +, −, *, /, ^ and parentheses. Use a dot for finite decimals. Exponents must be whole numbers from 0 to 12; −x^2 means −(x²), while (−x)^2 is positive x². Multiplication may be implicit in 2x, 2(x+1), x(x−1) and (x+1)(x−1); use * between numeric factors. Multiplication and division are processed from left to right: 1/2x means (1/2) × x. Use 1/(2x) only as the main division, with 1 in the dividend and 2x in the divisor.

Division inside an input is allowed only by a nonzero constant, such as x/2 or (x+1)/3. Equations, other variables, functions, negative or fractional powers and polynomial denominators inside an input are not supported. Limits per input: 2,000 characters, 512 tokens, 32 nested parentheses and 24 digits per numeric literal. Every intermediate polynomial must have degree at most 12; reduced rational coefficients are limited to 1,024 digits and the calculation to 50,000 work units. Expressions exceeding these limits keep your input and show an error.

All answers and steps use exact rational arithmetic. No rounded decimal is fed back into the calculation. This tool performs polynomial long division, not a root search, factorization or synthetic division.