The question is as follows:
How many possible rational roots are there for $2x^4 + 4x^3 - 6x^2 + 15x - 12 = 0$?
A. 4
B. 6
C. 8
D. 12
E. 16
I was taught that a polynomial of order $n$ has at most $n$ roots, hence A. However, the answer was E, and I don't have a clue what the answer is referring to.
Could someone please explain?
The question here is ambiguous. The answer given suggests that the context is the rational root theorem, and the implied question is "using the rational root theorem, how many rational numbers are possible roots of this equation". Obviously there are four complex roots, counted with multiplicity, which can be found or tested in various ways. Some of the roots may be rational. There are ways of testing how many real roots an equation has, for example, which would give a number $\le 4$.
The rational root theorem tells us that if a polynomial $p(x)$ of degree $n$ with integer coefficients has the form $p(x)=ax^n +\dots +b$, then any rational root $\frac rs$ in lowest terms will have $s|a$ and $r|b$. In particular if $a=\pm 1$ then any rational roots will be integers.
Here you have $n=4, a=2, b=12$. The possible values of $r$ are $\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12$ and the possible values of $s$ are $\pm 1, \pm 2$. Combining these gives the answer suggested.
The question seems therefore not to be asking about the roots once the equation is solved, but about the first stage in a process of seeking any rational roots. You have not given enough context for the question for me to know whether this is really valid, since the question as posed is ambiguous. But I would have taken the word "possible" in the question to indicate that something like the rational root theorem was involved, rather than a full and precise analysis of actual roots.