 In this video, we provide the solution to question number two for practice exam number two for math 1220, in which case we're given the rational function right here, x cube minus one over x squared times x squared plus two times x plus two quantity squared. And we're asked to determine what is the correct template for the partial fraction decomposition. Well, it is a proper fraction right because notice the denominator has two two and two. If you multiply it out the denominator would be degree six the numerator is degree three so it is a proper fraction so we don't have to worry about the numerator for the rest of this template here. So for each of the factors here, we need to get a partial fraction for all of the powers up to the one we have if it's repeated. So for example, since there's an x square here, we're going to need something like a over x and be over x squared. So this right here looks promising, right? So choices B and D have kind of what we're looking for. Likewise, we have this x plus two quantity squared. So I need to have a factor which has a x plus two and an x plus two squared. The numerators in all of these cases should just be a constant. So honestly D is looking like the best answer right now. I think with the information I have I could determine that because notice A doesn't have an A over x. C doesn't have an A over x squared. Same thing going on here. Now admittedly this right here, you could get away with that one because if you take this A x plus B that would simplify to something like this right here. But there's a problem with this one. We don't have the correct template for the x plus two squared. We would need the possibility of something like this. So A, C, E are all ruled out so far. Same thing for F, right? We don't have enough of the x squares or enough of the x plus two squared. So we can rule that out with B. We have the right template for the x squares. Not quite for this one. We have an x plus two. Then we have an x plus two squared. I mean this isn't even a proper function, a proper rational function there. And so we could rule out all the possibilities to get D just from what we've considered so far. But let's also look at the x squared plus two. That is an irreducible quadratic. So that is a situation where the numerator could be linear Cx plus D. You can see that none of the other ones have everything correct together. So D does have to be the correct template for this partial fraction decomposition.