I am doing some partial fraction decomposition and I came across this problem. Decompose $\frac{x-3}{x^2-7}$. How can this be done?
I was thinking that maybe you could do a difference of squares? But since 7 isn't a perfect square is this allowed? I looked all over the web and found nothing on this subject, every video and website talks about perfect squares, but what if they aren't perfect?
Following the pattern of perfect squares I can only assume the pattern is to do the following, in order to decompose: $\frac{A}{x-\sqrt{7}}+\frac{B}{x+\sqrt{7}}$
Is this correct?
Best Answer
Yes, you can do it, provided you don't mind using irrational real numbers. That is, if $a$ is positive, then you have that $$x^2-a=(x-\sqrt a)(x+\sqrt a).$$