1.

Resolve into partial fraction \( \frac{1}{(x-1)\left(x^{2}-9\right)} \)

Answer»

complex rational statement is decomposed into two or more simpler fractions using partial fractions. To divide fractions into several sub-fractions, partial fractions ideas are employed.

Given:

\frac{1}{(x-1)(x^{2}-9) }

Find

Resolve the expression to a partial fraction.

Solution:

The given expression can be written as,

\begin{gathered}\frac{1}{(x-1)(x^{2}-9) }= \frac{1}{(x-1)(x-3)(x+3) }\\\\ \frac{1}{(x-1)(x-3)(x+3) } = \frac{A}{(x-3)}+ \frac{B}{(x+3) }+ \frac{C}{(x-1) }\end{gathered}

\(\frac1{(x-1)(x^2-9)}\) = \(\frac1{(x-1)(x-3)(x+3)}\) = \(\frac A{(x-1)}+\frac {B}{x-3}+\frac C{x+3}\)

⇒ A (x - 3) (x + 3) + B (x - 1)(x + 3) + C(x - 1)(x - 3) = 1

Put x = 1, we get -8A = 1 ⇒ A = -1/8

Put x = 3, we get 12B  = 1 ⇒ B = 1/12

Put x = -3, we get 24C = 1 ⇒ C = 1/24

Hence, \(\frac1{(x-1)(x^2-9)}\)= \(\frac{1}{x-1}+\frac1{12}\frac1{x-3}+\frac{1}{24}\frac1{x+3}\)


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