Computation Question in the Ring of Polynomial with Integer Coefficients

[jmjlt88]

I have a quick question. The problem reads:

Prove that there is no integer m such that 3x²+4x + m is a factor of 6x⁴+50 in Z[x].

Now, Z[x] is not a field. So, the division algorithm for polynomials does not guarantee us a quotient and remainder. When I tried dividing 6x⁴+50 by 3x²+4x + m, it immediately would have push me out of the integers. So, 3x²+4x + m cannot be a factor no matter what m is.

My question, how can write that nicely in a proof? This may be a silly question. I have finished my first run through of Pinter's A Book of Abstract Algebra and I am now going back, re-writing proofs more concisely, fixing mistakes, and trying ones I skipped or did not "get." I only wrote an explanation similar to the one above for my answer with an attempt at divinding the polynomials.

 

[Hurkyl]

Based on your description, it sounds like you want to work in Q[x]. What does working in Q[x] tell you, and can you relate that fact to the thing you're trying to show?