Expert answers to "Write the partial fraction decomposition for the rational..."

osteoblogda

Answered question

2021-12-16

Write the partial fraction decomposition for the rational expression

\frac{3 x^{2} - 2 x - 1}{x {(x + 1)}^{2} {(x^{2} + 4)}^{2}}

censoratojk

Beginner2021-12-17Added 46 answers

Solving: Given

\frac{3 x^{2} - 2 x - 1}{x {(x + 1)}^{2} {(x^{2} + 4)}^{2}}

Write the partial fraction decomposition for the rational expression
Using partial fraction rule

\sqrt{\frac{a x + b}{{(x + a)}^{2} (x + b)}} = \frac{A}{(x + a)} + \frac{B}{{(x + a)}^{2}} + \frac{C}{(x + b)}

\sqrt{a x + b} {x^{2} + a^{2}) (x + b)}} = \frac{A x + b}{x^{2} + a^{2}} + \frac{C}{(x + b)}

Step 2
Then for are given

\frac{3 x^{2} - 2 x - 1}{x {(x + 1)}^{2} {(x^{2} + 4)}^{2}}

\frac{3 x^{2} - 2 x - 1}{x {(x + 1)}^{2} {(x^{2} + 4)}^{2}} = \frac{A}{x} + \frac{B}{(x + 1)} + \frac{C}{{(x + 1)}^{2}} + \frac{D x + E}{(x^{2} + 4)} + \frac{F x + G}{{(x^{2} + 4)}^{2}}

Hence write the partial fraction dicompositien for the rational expression

\frac{3 x^{2} - 2 x - 1}{x {(x + 1)}^{2} {(x^{2} + 4)}^{2}} = \frac{A}{x} + \frac{B}{(x + 1)} + \frac{C}{{(x + 1)}^{2}} + \frac{D x + E}{(x^{2} + 4)} + \frac{F x + G}{{(x^{2} + 4)}^{2}}

Dawn Neal

Beginner2021-12-18Added 35 answers

Step 1
Given:

\frac{3 x^{2} - 2 x - 1}{x {(x + 1)}^{2} {(x^{2} + 4)}^{2}}

{(x + 1)}^{2} = x^{2} + 2 x + 1

= \frac{3 x^{2} - 2 x - 1}{x (x^{2} + 2 x + 1) {(x^{2} + 4)}^{2}}

{(x^{2} + 4)}^{2} = x^{4} + 8 x^{2} + 16

\frac{3 x^{2} - 2 x - 1}{x (x^{2} + 2 x + 1) (x^{4} + 8 x^{2} + 16)}

Expand

(x^{2} + 2 x + 1) (x^{4} + 8 x^{2} + 16) : x^{7} + 2 x^{6} + 9 x^{5} + 16 x^{4} + 24 x^{3} + 32 x^{2} + 16 x

= \frac{3 x^{2} - 2 x - 1}{x^{7} + 2 x^{6} + 9 x^{5} + 16 x^{4} + 24 x^{3} + 32 x^{2} + 16 x}

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