"How do I use the partial fractions technique..." was answered by our community

Efrain Watkins

Answered question

2022-04-15

How do I use the partial fractions technique in this case?

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

unduncjineei5r3

Beginner2022-04-16Added 19 answers

The partial fraction decomposition takes the form

F (x) = \frac{x - 1}{(x^{2} - x + 1) (x + 1)} = \frac{a}{x + 1} + \frac{b x + c}{x^{2} - x + 1}

and notice that

a = F (x) (x + 1) ∣_{x = - 1} = - \frac{2}{3}

and

0 = lim_{x \to \infty} x F (x) = a + b \Rightarrow b = \frac{2}{3}

and finaly

F (0) = - 1 = a + c \Rightarrow c = - \frac{1}{3}

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