`(8x - 12)/(x^2(x^2 + 2)^2)` Write the partial fraction decomposition of the rational expression. Check your result algebraically.

You need to decompose the fraction in simple irreducible fractions, such that:

`(8x-12)/(x^2(x^2+2)^2) = A/x + B/(x^2) + (Cx+D)/(x^2+2) + (Ex+F)/((x^2+2)^2)`

You need to bring to the same denominator all fractions, such that:

`8x - 12 = Ax(x^2+2)^2 + B(x^2+2)^2 + (Cx+D)*x^2*(x^2+2) + (Ex+F)*x^2`

`8x - 12 = Ax(x^4 + 4x^2 + 4) + Bx^4 + 4Bx^2 + 4B + (Cx^3 + Dx^2)*(x^2+2) + Ex^3 + Fx^2`

`8x - 12 = Ax^5 + 4Ax^3...

You need to decompose the fraction in simple irreducible fractions, such that:

`(8x-12)/(x^2(x^2+2)^2) = A/x + B/(x^2) + (Cx+D)/(x^2+2) + (Ex+F)/((x^2+2)^2)`

You need to bring to the same denominator all fractions, such that:

`8x - 12 = Ax(x^2+2)^2 + B(x^2+2)^2 + (Cx+D)*x^2*(x^2+2) + (Ex+F)*x^2`

`8x - 12 = Ax(x^4 + 4x^2 + 4) + Bx^4 + 4Bx^2 + 4B + (Cx^3 + Dx^2)*(x^2+2) + Ex^3 + Fx^2`

`8x - 12 = Ax^5 + 4Ax^3 + 4Ax + Bx^4 + 4Bx^2 + 4B + Cx^5 + 2Cx^3 + Dx^4 + 2Dx^2 + Ex^3 + Fx^2`

You need to group the terms having the same power of x:

`8x - 12 = x^5(A + C) + x^4(B+D) + x^3(4A + 2C + E) + x^2(4B + 2D + F) + x(4A) + + 4B`

Comparing the expressions both sides yields:

`A + C = 0 => A =-C`

`B+D = 0 => B = - D`

`4A + 2C + E = 0 => 2A + E = 0 => E = -2A`

`4B + 2D + F = 0 => F = -2D`

`4A = 8 => A = 2 => C = -2 => E = -4`

`4B=-12 => B = -3 => D = 3 => F = -6`

Hence, decomposing the fraction, yields `(8x-12)/(x^2(x^2+2)^2) = 2/x- 3/(x^2) + (-2x+3)/(x^2+2) + (-4x-6)/((x^2+2)^2).`