Question

How do you write the partial fraction decomposition of the rational expression `(5x^2+7x-4)/(x^3+4x^2)`?

Answer 1

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

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

so, `5x^2 + 7x - 4 = (Ax + B)(x + 4) + Cx^2`

Setting x = 0 we have:

-4 = 4B

i.e. B = -1

With x = -4 we have:

48 = 16C

i.e. C = 3

With x = 1 we have:

8 = 5(A + B) + C

=> 8 = 5A - 5 + 3

so, 5A = 10

i.e. A = 2

Hence, `(5x^2 + 7x - 4)/(x^3 + 4x^2) = (2x - 1)/x^2 + 3/(x + 4)`

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