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Algebra readiness · Choose and solve partial fractions

Build the template before finding coefficients

Factor types determine the terms you need. A denominator can contain more than one type, so combine the relevant rules.

  1. Check degrees, then factor

    Divide first if the numerator degree is at least the denominator degree. Keep the polynomial part, then decompose the proper remainder over real factors.

    deg⁡P<deg⁡Q\deg P<\deg Q for the fraction being decomposed.

    Simple linear factorOne constant numerator

    For a factor ax+bax+b, use A/(ax+b)A/(ax+b), with a≠0a\ne0. Give distinct factors their own coefficients.

    Repeated linear factorInclude every power

    For (ax+b)m(ax+b)^m, include A1/(ax+b)+⋯+Am/(ax+b)mA_1/(ax+b)+\cdots+A_m/(ax+b)^m. Missing a lower power leaves an incomplete template.

    Non-repeated irreducible quadraticAllow a linear numerator

    For a quadratic q(x)q(x) with no real roots, use (Bx+C)/q(x)(Bx+C)/q(x). A constant numerator alone may not suffice.

    Clear denominators and solve the polynomial identity by substitution or coefficient comparison. Recombine to check. Keep the original excluded inputs even though you may substitute those values into the polynomial identity.