Partial fractions

Algebra

A-level Maths (9MA0)

ORIGINAL

One fraction per linear factor, then substitute the clever values of x to pick off each constant.

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Question

Express

\dfrac{3x + 1}{(x - 1)(x + 2)}

in partial fractions.

Every step worked, with the reasoning.

1
$\dfrac{3x+1}{(x-1)(x+2)} \equiv \dfrac{A}{x-1} + \dfrac{B}{x+2}$
Set up one fraction per linear factor.
2
$3x + 1 \equiv A(x+2) + B(x-1)$
Multiply both sides by $(x-1)(x+2)$ .
3
$x = 1:\ \ 4 = 3A \Rightarrow A = \dfrac{4}{3}$
Substitute $x = 1$ to kill the $B$ term.
4
$x = -2:\ \ -5 = -3B \Rightarrow B = \dfrac{5}{3}$
Substitute $x = -2$ to kill the $A$ term.
5
$\dfrac{3x+1}{(x-1)(x+2)} \equiv \dfrac{4}{3(x-1)} + \dfrac{5}{3(x+2)}$
Write out the result.

Answer: $\dfrac{4}{3(x-1)} + \dfrac{5}{3(x+2)}$