Quote:
Originally Posted by R.D. Silverman
Firstly, it does not matter what "everyone thinks". If this proof is
being done by a student, then the only thing that matters is what the
teacher will accept. If it appears in a publication, then what matters is
what professional mathematicians will accept.
As a *student*, what you have done does not prove the result.
You would indeed need to either add the proofs of (1), (2), (3), or
refer to a previously established result. And when I say "established
result", I mean a result established as part of the course. This last bit
depends a bit on the level of the course. A student in an upperlevel
undergrad course can point to a known, published result that is not
part of the course itself. But in a lowerdivision course, I doubt whether
a teacher would accept external references.
Also, if this were done as a proof in a precalculus class where limits
and the intermediate value theorem were not yet established, this proof
would not be accepted by the teacher. Generally, one is not allowed to
use a more "advanced" method to prove an elementary result. The teacher
would expect a proof using results from the course or previous courses.
Of course, if given as a homework problem, then the teacher should
specify "prove all claims" as part of the problem statement.

An omission on my part:
Even if this proof were done in a calculus course, you would still need to
refer to the intermediate value theorem to establish the claim that the
polynomial does indeed cross the axis. Merely stating that it crosses
the axis would be insufficient.