How this calculator works
Polynomial division first separates an improper fraction into a polynomial plus a proper fraction. Each power of a linear factor needs a constant numerator, and each power of an irreducible quadratic needs a linear numerator. Clearing denominators gives a linear system for these coefficients. Supply the real factorization explicitly; for a non-monic denominator, divide the numerator by its overall leading multiplier first.
Formula
P/Q = S + R/Q. For (x-r)^m use sum A_j/(x-r)^j. For an irreducible q(x)^m use sum (B_j*x+C_j)/q(x)^j, for j = 1,...,m.
Worked example
For 3x/((x + 1)(x - 2)), clearing denominators gives 3x = A(x - 2) + B(x + 1). Comparing coefficients gives A = 1 and B = 2. Thus the result is 1/(x + 1) + 2/(x - 2).
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Frequently asked questions
Can factors repeat?
Yes. Enter each distinct factor once and give its positive whole-number multiplicity. Every denominator power up to that multiplicity is included.
Why enter quadratic factors separately?
A real irreducible quadratic has no real roots, so it cannot be represented by a list of real linear roots.
Are poles removed by decomposition?
No. The original denominator restrictions still apply, including when a factor cancels algebraically.