Basic Partial Fraction Decomposition

00:00 5 marks

Decompose the rational function \( f(x)=\frac{2x+1}{(x-1)(x+2)} \) into partial fractions. Subsequently, verify that your decomposition is correct by recombining the fractions to obtain the original expression.

Part a

5 minutes 3 marks

Question

Decompose the rational function \( f(x)=\frac{2x+1}{(x-1)(x+2)} \) into partial fractions.

Hint Hint

Hint

Assume a form \( \frac{A}{x-1} + \frac{B}{x+2} \) and determine \(A\) and \(B\) by equating coefficients.

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Complete Answer

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

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Part b

3 minutes 2 marks

Question

Verify that your decomposition from part (a) is correct by recombining the fractions to obtain the original expression.

Hint Hint

Hint

Add the two partial fractions obtained in part (a) and simplify the result.

Answer Answer
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Answer Answer

Complete Answer

Recombining, \(\frac{1}{x-1}+\frac{1}{x+2} = \frac{(x+2)+(x-1)}{(x-1)(x+2)} = \frac{2x+1}{(x-1)(x+2)}\).

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