How this calculator works
Long division cancels the highest remaining power in the dividend at each step. Divide the leading terms, subtract that multiple of the divisor, and continue until the remainder has lower degree than the divisor. If the dividend degree is already smaller, the quotient is zero and the dividend is the remainder. Decimal coefficients can leave small numerical residuals.
Formula
P(x) = Q(x)S(x) + R(x), where R = 0 or degree(R) < degree(Q). Thus P/Q = S + R/Q wherever Q is non-zero.
Worked example
Dividing x^2 - 3x + 2 by x - 1 gives quotient x - 2 and remainder 0. Dividing x^2 + 1 by x + 1 gives quotient x - 1 and remainder 2.
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Frequently asked questions
Can I divide by the zero polynomial?
No. At least one divisor coefficient must be non-zero.
What if the divisor is constant?
Every dividend coefficient is divided by that constant, and the remainder is zero.
Does zero remainder mean every x is allowed in the fraction?
No. Zeros of the original divisor remain excluded from the original rational expression.