Question

How do you express `(5x^2+7x-4)/(x^3+4x^2)` in partial fractions?

Answer 1

`color(red)((5x^2+7x-4)/(x^3+4x^2)=(-1)/x^2+2/x+3/(x+4))`

Explanation:

We start from the denominator of the rational expression

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

We will use 3 variables here namely A, B, and C.
Determine the LCD`=x^2(x+4)`

`(5x^2+7x-4)/(x^2(x+4))=A/x^2+B/x+C/(x+4)`

`(5x^2+7x-4)/(x^2(x+4))=(A(x+4)+B(x^2+4x)+Cx^2)/(x^2(x+4))`

Arrange the coefficients of `x^2, x^1, x^0`

`(5x^2+7x^1-4x^0)/(x^2(x+4))=((B+C)x^2+(A+4B)x^1+4A*x^0)/(x^2(x+4))`

Now we can set up the equations

`B+C=5`
`A+4B=7`
`4A=-4`

Simultaneous solution of these equations result to

`A=-1` and `B=2` and `C=3`

and it follows

`(5x^2+7x-4)/(x^2(x+4))=A/x^2+B/x+C/(x+4)`

`color(blue)((5x^2+7x-4)/(x^2(x+4))=(-1)/x^2+2/x+3/(x+4))`

God bless....I hope the explanation is useful.