Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2012 / Paper 1 / 6 / Solution

A repeated linear factor requires both reciprocal powers. Write the partial fraction decomposition as

\frac{A}{x - 1} + \frac{B}{( x - 1 ) ^{2}} + \frac{C}{x + 2} .

Multiplication by the common denominator gives

4 x^{2} - x = A (x - 1) (x + 2) + B (x + 2) + C (x - 1)^{2}

. Setting

x = 1

gives

B = 1

, and setting

x = - 2

gives

C = 2

. Comparing the coefficients of

x^{2}

then gives

A = 2

. Thus

\frac{4 x ^{2} - x}{( x - 1 ) ^{2} ( x + 2 )} = \frac{2}{x - 1} + \frac{1}{( x - 1 ) ^{2}} + \frac{2}{x + 2}, x \neq = 1, - 2.

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