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

1 Answer
George C.
Sep 10, 2016

(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)

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