Question

How do you write the partial fraction decomposition of the rational expression `(6x+5) / (x+2) ^4`?

Answer 1

Start with:

`(6x+5)/(x+2)^4 = A/(x+2)^4 + B/(x+2)^3+C/(x+2)^2+D/(x+2)`

Multiply both sides by `(x+2)^4`:

`6x+5 = A + B(x+2)+C(x+2)^2+D(x+2)^3`

Let x = -2

`6(-2)+5 = A`

`A = -7`

Add 7 to both sides of equation:

`6x+12 = B(x+2)+C(x+2)^2+D(x+2)^3`

Let x = 0:

`[ (2,4,8,|,12), ]`

Let x = -1

`[ (2,4,8,|,12), (1,1,1,|,6) ]`

Let x = 1

`[ (2,4,8,|,12), (1,1,1,|,6), (3,9,27,|,18) ]`

`R_1harrR_2`

`[ (1,1,1,|,6), (2,4,8,|,12), (3,9,27,|,18) ]`

`R_2-2R_1toR_2`

`[ (1,1,1,|,6), (0,2,6,|,0), (3,9,27,|,18) ]`

`R_3-3R_1toR_3`

`[ (1,1,1,|,6), (0,2,6,|,0), (0,6,24,|,0) ]`

`R_2/2toR_2`

`[ (1,1,1,|,6), (0,1,3,|,0), (0,6,24,|,0) ]`

`R_3/6toR_3`

`[ (1,1,1,|,6), (0,1,3,|,0), (0,1,4,|,0) ]`

`R_3-R_3toR_3`

`[ (1,1,1,|,6), (0,1,3,|,0), (0,0,1,|,0) ]`

It is obvious that C and D are 0 so we can clear the matrix:

`[ (1,0,0,|,6), (0,1,0,|,0), (0,0,1,|,0) ]`

`B = 6 and C = D = 0`

`(6x+5)/(x+2)^4 = -7/(x+2)^4 + 6/(x+2)^3`