`(x^2 + 12x + 12)/(x^3 - 4x)` Write the partial fraction decomposition of the rational expression. Check your result algebraically.
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`[x^2+12x+12]/[x^3-4x]=[x^2+12x+12]/[x(x+2)(x-2)]`
`[x^2+12x+12]/(x^3-4x)=A/x+B/(x+2)+C/(x-2)`
Multiply through by the LCD `x^3+4x.`
`x^2+12x+12=A(x^2-4)+Bx(x-2)+Cx(x+2)`
`x^2+12x+12=Ax^2-4A+Bx^2-2Bx+Cx^2+2Cx`
`x^2+12x+12=(A+B+C)x^2+(-2B+2C)x+(-4A)`
Equate coefficient of like terms. Then solve for A, B, and C.
`1=A+B+C`
`12=-2B+2C`
`12=-4A`
`A=-3`
`1=A+B+C`
`1=-3+B+C`
`4=B+C`
Solve for B and C using the elimination method.
`4=B+C`
`12=-2B+2C`
Multiply the first equation by 2. Then solve using the elimination method.
`8=2B+2C`
`12=-2B+2C`
_________________
20=4C
`C=5`
`1=A+B+C`
`1=-3+B+5`
`B=-1`
`A=-3, B=-1, C=5`
`(x^2+12x+12)/(x^3-4x)=-3/x-1/(x+2)+5/(x-2)`
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