# Write the partial fraction decomposition for the rational expression \frac{3x^{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
$\frac{3{x}^{2}-2x-1}{x{\left(x+1\right)}^{2}{\left({x}^{2}+4\right)}^{2}}$
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## Expert Answer

censoratojk
Answered 2021-12-17 Author has 46 answers
Solving: Given $\frac{3{x}^{2}-2x-1}{x{\left(x+1\right)}^{2}{\left({x}^{2}+4\right)}^{2}}$
Write the partial fraction decomposition for the rational expression
Using partial fraction rule
$\sqrt{\frac{ax+b}{{\left(x+a\right)}^{2}\left(x+b\right)}}=\frac{A}{\left(x+a\right)}+\frac{B}{{\left(x+a\right)}^{2}}+\frac{C}{\left(x+b\right)}$
$\sqrt{ax+b}\left\{{x}^{2}+{a}^{2}\right)\left(x+b\right)\right\}\right\}=\frac{Ax+b}{{x}^{2}+{a}^{2}}+\frac{C}{\left(x+b\right)}$
Step 2
Then for are given $\frac{3{x}^{2}-2x-1}{x{\left(x+1\right)}^{2}{\left({x}^{2}+4\right)}^{2}}$
$\frac{3{x}^{2}-2x-1}{x{\left(x+1\right)}^{2}{\left({x}^{2}+4\right)}^{2}}=\frac{A}{x}+\frac{B}{\left(x+1\right)}+\frac{C}{{\left(x+1\right)}^{2}}+\frac{Dx+E}{\left({x}^{2}+4\right)}+\frac{Fx+G}{{\left({x}^{2}+4\right)}^{2}}$
Hence write the partial fraction dicompositien for the rational expression
$\frac{3{x}^{2}-2x-1}{x{\left(x+1\right)}^{2}{\left({x}^{2}+4\right)}^{2}}=\frac{A}{x}+\frac{B}{\left(x+1\right)}+\frac{C}{{\left(x+1\right)}^{2}}+\frac{Dx+E}{\left({x}^{2}+4\right)}+\frac{Fx+G}{{\left({x}^{2}+4\right)}^{2}}$
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Dawn Neal
Answered 2021-12-18 Author has 35 answers
Step 1
Given:
$\frac{3{x}^{2}-2x-1}{x{\left(x+1\right)}^{2}{\left({x}^{2}+4\right)}^{2}}$
${\left(x+1\right)}^{2}={x}^{2}+2x+1$
$=\frac{3{x}^{2}-2x-1}{x\left({x}^{2}+2x+1\right){\left({x}^{2}+4\right)}^{2}}$
${\left({x}^{2}+4\right)}^{2}={x}^{4}+8{x}^{2}+16$
$\frac{3{x}^{2}-2x-1}{x\left({x}^{2}+2x+1\right)\left({x}^{4}+8{x}^{2}+16\right)}$
Expand
$\left({x}^{2}+2x+1\right)\left({x}^{4}+8{x}^{2}+16\right):\phantom{\rule{1em}{0ex}}{x}^{7}+2{x}^{6}+9{x}^{5}+16{x}^{4}+24{x}^{3}+32{x}^{2}+16x$
$=\frac{3{x}^{2}-2x-1}{{x}^{7}+2{x}^{6}+9{x}^{5}+16{x}^{4}+24{x}^{3}+32{x}^{2}+16x}$
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