Write out the form of the partial fraction decomposition of the function. Do not determine the numerical value the coefficient.

Question

(x^2/(x^2+x-56)) =

I tried solving this question by dividing the numerator by the denominator and I got 1+ (56-x/(x^2+x-56). I simplified the denominator and I got 1+(56-x/(x+8)(x-7) ). Lastly, I got the answer A/x-7 plus B/x+8 . But unfortunately my answer was incorrect. I wanted to know where I went wrong and how could I correctly solve this question. Thank you in advance!

Answer 1

Hi Rozhan,

Although the original polynomial division you performed is correct, it is largely unnecessary to complete partial fraction decomposition. If we look at the answer you arrived at that was incorrect, A/(x-7) + B/(x+8), where A and B are constants, you can see that if you seek to get common denominators to combine the fractions, you would get the following:

[A(x + 8) + B(x - 7)] / [(x - 7)(x + 8)]

What we should notice here is that there is no way to arrive at an x² term in the numerator. Thus, the setup you want to use for the partial fraction decomposition is to use something like this:

(Ax + B) / (x - 7) + (Cx + D) / (x + 8)

When setting up the numerators for partial fraction decomposition, you always use an expression that is one degree less than the degree of the original denominator. So, because the degree of the original denominator is 2, you need to have each numerator be a linear expression (degree 1).

Hope that helps! Let me know if you have any questions.