The Examiners’ Report indicates that about half of the students gave the intended answer of D, with about a third giving the incorrect answer B. The Report notes:
Option B did not account for common factors and its last term is not irreducible, so should not have Dx in the numerator.
The worst kind of exam question is one that rewards mindless button-pushing and actively punishes intelligent consideration. The above question is of the worst kind. It is also pointless, nasty and self-trippingly overcute.
As John points out in the comments, the question can simply be done by pressing CAS buttons. But, alternatively, the question also just appears to require, and to invite, a simple understanding of partial fraction form. Which brings us to the nastiness: the expected partial fraction form is not a listed option.
So, what to make of it? Not surprisingly, many students opted for B, the superficially most plausible answer. A silly mistake, you silly, silly student! You shoulda just listened to your teacher and pushed the fucking buttons.
The trick, of course, is that the numerator factorises, cancelling with the denominator and leading to the intended answer, D. The problem with the trick is that it is antimathematical and wrong:
- As Damo notes, the original rational function is undefined at x = -1, which is lost in the intended answer.
- As Damo also points out, there is no transparent, non-computational way to check that the coefficients in answer D would, as demanded by the question, be non-zero.
- It is not standard or particularly natural to hunt for common factors before breaking into partial fractions. Any such factors will anyway become apparent in the partial fractions.
- To refer to the partial fraction form is actively misleading. Though partial fraction decomposition can be defined so as to be unique, in practice it is usually not helpful to do so, and the VCE Study Design never does so. In particular, if answer B had contained a final numerator of Dx + E then this answer would be valid and, in certain contexts, natural and useful.
- The examiners’ comment on answer B is partly wrong and partly incomprehensible. One can pedantically object to the reducible denominator but if that is the objection then why whine about the Dx in the numerator? And yes, answer B is missing the constant E, which in general is required, and happens to be required for the given rational function. For a specific rational function, however, one might have E = 0. Which brings us back to Damo’s point, that without actually computing the partial fractions there is no way of determining whether answer B is valid.
But of course all that is way, way too much to think about in a speed-test exam. Much better to just listen to your teacher and push the fucking buttons.