Homework Help: Computation Question in the Ring of Polynomial with Integer Coefficients

1. Jul 29, 2012

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.

2. Jul 29, 2012

Hurkyl

Staff Emeritus
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?