Partial Fractions

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Partial Fractions: Undo the Combine

\frac{5x+1}{(x-1)(x+2)} = \frac{A}{x-1} + \frac{B}{x+2}

Splitting is just combining fractions in reverse. Reflex: Linear factor in denominator → one constant over each factor.

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Partial Fractions

6 worked examples

Partial Fractions: Undo the Combine

\frac{5x+1}{(x-1)(x+2)} = \frac{A}{x-1} + \frac{B}{x+2}

Splitting is just combining fractions in reverse. Reflex: Linear factor in denominator → one constant over each factor.

Factorize the Denominator First

\begin{gathered}\frac{3x+1}{(x+2)(x^2+4x+3)}\\[3pt] = \frac{3x+1}{(x+2)(x+3)(x+1)}\\[3pt] = \frac{A}{x+2} + \frac{B}{x+3} + \frac{C}{x+1}\end{gathered}

Three linear factors → three constants on top. Reflex: Always try to factorize the quadratic before you choose the split.

Unfactorable Quadratic → Bx+C on Top

\begin{gathered}\frac{x+5}{(x+2)(x^2+x+1)}\\[3pt] = \frac{A}{x+2} + \frac{Bx+C}{x^2+x+1}\end{gathered}

Irreducible quadratic stays whole underneath — its numerator must be linear, not constant. Reflex: Cannot factorize the quadratic → $Bx+C$ on top, never just $B$ .

Repeated Linear Factor: Split Twice

\begin{gathered}\frac{5}{(x+2)(x-1)^2}\\[3pt] = \frac{A}{x+2} + \frac{B}{x-1} + \frac{C}{(x-1)^2}\end{gathered}

A squared factor needs BOTH a plain and a squared denominator term. Reflex: One squared factor in the question → two fractions for it in the split.

Improper Fraction? Long Division First

\begin{gathered}\frac{4x^3+7x^2-13x-2}{(x+3)(x-1)}\\[3pt] = \underbrace{4x-1}_{\text{quotient}} + \underbrace{\frac{x-5}{(x+3)(x-1)}}_{\text{proper}}\end{gathered}

Top degree < bottom → proper, split straight away. Top degree ≥ bottom → improper, long-divide first. Reflex: Check degrees before doing anything else.

Which Split? Match the Factor

\begin{gathered}(x-a) \to \frac{A}{x-a}\\[3pt] (x-a)^2 \to \frac{A}{x-a}+\frac{B}{(x-a)^2}\\[3pt] (x^2+c) \to \frac{Ax+B}{x^2+c}\end{gathered}

Improper → divide first, then factorize the denominator fully and read its factors one at a time. Reflex: The denominator's factors dictate the form.

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