Write the partial fraction decomposition for the rational expression

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

osteoblogda
2021-12-16
Answered

Write the partial fraction decomposition for the rational expression

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

You can still ask an expert for help

censoratojk

Answered 2021-12-17
Author has **46** answers

Solving: Given $\frac{3{x}^{2}-2x-1}{x{(x+1)}^{2}{({x}^{2}+4)}^{2}}$

Write the partial fraction decomposition for the rational expression

Using partial fraction rule

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

$\sqrt{ax+b}\{{x}^{2}+{a}^{2})(x+b)\}\}=\frac{Ax+b}{{x}^{2}+{a}^{2}}+\frac{C}{(x+b)}$

Step 2

Then for are given$\frac{3{x}^{2}-2x-1}{x{(x+1)}^{2}{({x}^{2}+4)}^{2}}$

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

Hence write the partial fraction dicompositien for the rational expression

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

Write the partial fraction decomposition for the rational expression

Using partial fraction rule

Step 2

Then for are given

Hence write the partial fraction dicompositien for the rational expression

Dawn Neal

Answered 2021-12-18
Author has **35** answers

Step 1

Given:

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

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

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

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

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

Expand

$({x}^{2}+2x+1)({x}^{4}+8{x}^{2}+16):{\textstyle \phantom{\rule{1em}{0ex}}}{x}^{7}+2{x}^{6}+9{x}^{5}+16{x}^{4}+24{x}^{3}+32{x}^{2}+16x$

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

Given:

Expand

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