How do you write the partial fraction decomposition of the rational expression $(x^2 -111)/ (x^4- x^2- 72)$ ?

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Answer 1 · 2015-12-15T02:33:39

$20/(17(x-3))-20/(17(x+3))-103/(17(x^2+8)$

Factor the denominator.

$x^4-x^2-72=(x^2-9)(x^2+8)=(x+3)(x-3)(x^2+8)$

Write the partial fraction decomposition expression.

$(x^2-111)/((x+3)(x-3)(x^2+8))=A/(x+3)+B/(x-3)+(Cx+D)/(x^2+8)$

$x^2-111=A(x-3)(x^2+8)+B(x+3)(x^2+8)+(Cx+D)(x^2-9)$

$x^2-111=A(x^3-3x^2+8x-24)+B(x^3+3x^2+8x+24)+Cx^3-9Cx+Dx^2-9D$

$x^2-111=x^3(A+B+C)+x^2(-3A+3B+D)+x(8A+8B-9C)+1(-24A+24B-9D)$

The following system can be deduced:

${(A+B+C=0),(-3A+3B+D=1),(8A+8B-9C=0),(-24A+24B-9D=111):}$

Solve the system:

${(A=-20/17),(B=20/17),(C=0),(D=-103/17):}$

Plug in these values:

$(x^2 -111)/ (x^4- x^2- 72)=20/(17(x-3))-20/(17(x+3))-103/(17(x^2+8)$

How do you write the partial fraction decomposition of the rational expression (x^2 -111)/ (x^4- x^2- 72) (x^2 -111)/ (x^4- x^2- 72)?