Practice Exercises

Partial Fractions

1. Express the following in partial fractions;

\begin{matrix}(a) 3 \frac{1}{(x + 1) (x − 1)} \\ (b) & \frac{x}{(x − 4) (x − 1)} \end{matrix}  \begin{matrix}(c) & \frac{x − 1}{(x + 2) (x − 2)} \\ (d) 2 \frac{1}{(2 x − 1) (x − 2)} \end{matrix}  \begin{matrix}(e) & \frac{x + 3}{x (x + 1)} \\ (f) & \frac{2 x − 1}{(x + 1) (3 x + 2)} \end{matrix}
2. (a) 3 \frac{x}{(x − 1) (x − 2) (x − 3)}  (b) \frac{x^{2} − 2 x + 4}{x (x − 3) (x + 1)}
\begin{matrix}3. (a) 2 \frac{1}{(x − 1) (x^{2} + 1)} \\ (b) & \frac{x − 3}{(x + 4) (x^{2} − 2)} \end{matrix}  \begin{matrix}(c) & \frac{x + 3}{x (x^{2} + 2)} \\ (d) & \frac{2 x^{2} + x + 1}{(x − 3) (2 x^{2} − 1)} \end{matrix}  \begin{matrix}(e) & \frac{x^{2} + 1}{x (2 x^{2} − 1) (x − 1)} \\ (f) & \frac{2 x^{2} + 1}{(x^{2} + 1) (x − 1)} \end{matrix}

4. Convert the following expression to partial fractions.

\frac{x^{3} − 1}{(x + 2) (1 + 2 x) (x^{2} + 1)}

5. By factorising the denominator write the following expression in partial fractions.

\frac{x^{2} + x + 1}{(x^{2} − 1) (x^{2} + 1)}

6. Express the following in partial fractions;

\begin{matrix}(a) & \frac{x}{(x − 1) (x − 2)^{2}} \\ (b) & \frac{x^{2} − 1}{x^{2} (2 x + 2)} \end{matrix}  \begin{matrix}(c) 3 \frac{1}{x (3 x − 1)^{2}} \\ (d) & \frac{1}{(x + 2) (x − 1)^{2}} \end{matrix}  \begin{matrix}(e) & \frac{1}{(x − 2) (x + 3)^{2}} \\ (f) & \frac{x − 2}{(3 x + 1) (x − 1)^{2}} \end{matrix}

7. Given that

\frac{x^{2} − 2}{(x + 3) (x − 1)} = A + \frac{B}{x + 3} + \frac{C}{x − 1}

find the values of the constants A, B and C.

8. Given that

\frac{x^{2} − x − 24}{(x + 2) (x − 4)} = A + \frac{B}{x + 2} + \frac{C}{x − 4}

find the values of the constants A, B and C.

Answers

1. (a) −3/2 (x + 1)^{−1} + 3/2 (x − 1)^{−1} ; (b) 4/3 (x − 4)^{−1} − 1/3 (x − 1)^{−1} ; (c) 3/4 (x + 2)^{−1} + 1/4 (x − 2)^{−1}  (d) −4/3 (2 x − 1)^{−1} + 2/3 (x − 2)^{−1} ; (e) 3 x^{−1} − 2 (x + 1)^{−1} ; (f) 3 (x + 1)^{−1} − 7 (3 x + 2)^{−1}

2. (a) 3/2 (x − 1)^{−1} − 6 (x − 2)^{−1} + 9/2 (x − 3)^{−1}

(b) −4/3 x^{−1} + \frac{7}{12} (x − 3)^{−1} + 7/4 (x + 1)^{−1}

3. (a) (x − 1)^{−1} − \frac{x + 1}{x^{2} + 1} ; (b) −1/2 (x + 4)^{−1} + 1/2 \frac{x − 2}{x^{2} − 2} ; (c) 3/2 x^{−1} − 1/2 \frac{−2 + 3 x}{x^{2} + 2}  (d) \frac{22}{17} (x − 3)^{−1} − 1/17 \frac{13 + 10 x}{2 x^{2} − 1} ; (e) x^{−1} + 2 (x − 1)^{−1} − 3 \frac{1 + 2 x}{2 x^{2} − 1} ; (f) 3/2 (x − 1)^{−1} + 1/2 \frac{x + 1}{x^{2} + 1}

4. 3/5 (x + 2)^{−1} − 3/5 (1 + 2 x)^{−1} + 1/5 \frac{x − 1}{x^{2} + 1}

5. 3/4 (x − 1)^{−1} − 1/4 (x + 1)^{−1} − 1/2 \frac{x}{x^{2} + 1}

6. (a) (x − 1)^{−1} + 2 (x − 2)^{−2} − (x − 2)^{−1} (b) −1/2 x^{−2} + 1/2 x^{−1}

(c) 3 x^{−1} + 9 (3 x − 1)^{−2} − 9 (3 x − 1)^{−1}

(d) 1/9 (x + 2)^{−1} + 1/3 (x − 1)^{−2} − 1/9 (x − 1)^{−1}

(e) 1/25 (x − 2)^{−1} − 1/5 (x + 3)^{−2} − 1/25 (x + 3)^{−1}

(f) −\frac{21}{16} (3 x + 1)^{−1} − 1/4 (x − 1)^{−2} + \frac{7}{16} (x − 1)^{−1}

7. 1 − 7/4 (x + 3)^{−1} − 1/4 (x − 1)^{−1}

8. 1 + 3 (x + 2)^{−1} − 2 (x − 4)^{−1}

August 14, 2001