Question

How do you write the partial fraction decomposition of the rational expression `(9x^2+9x+40)/(x(x^2+5))`?

Answer 1

`(9x^2+9x+40)/(x(x^2+5)) = 8/x+(9x+1)/(x^2+5)`

Explanation:

`(9x^2+9x+40)/(x(x^2+5)) = A/x+(Bx+C)/(x^2+5)`

`color(white)((9x^2+9x+40)/(x(x^2+5))) = (A(x+5)+(Bx+C)x)/(x(x^2+5))`

`color(white)((9x^2+9x+40)/(x(x^2+5))) = (Bx^2+(A+C)x+5A)/(x(x^2+5))`

Equating coefficients, we find:

`{ (B=9), (A+C=9), (5A=40) :}`

Hence:

`{ (A=8), (B=9), (C=1) :}`

So:

`(9x^2+9x+40)/(x(x^2+5)) = 8/x+(9x+1)/(x^2+5)`