(adsbygoogle = window.adsbygoogle || []).push({}); 3. Show that the polynomial [tex]p(x) = x^4 + 5x^2 + 3x + 2[/tex] is irreducible in [tex] \bbmath{Q} [x] [/tex].

This on has me totally stumped. I've seen many other problems on polynomial reducibility, but they are all solvable by Eisenstein, or by checking for irreducibility in a quotient ring by a proper ideal. Neither applies here.

I know that if it does reduce, then there are only two cases to check.

Case 1: [tex]p(x) = a(x)b(x)[/tex] where [tex]deg(a(x)) = 3, deg(b(x)) = 1[/tex].

This case is easy to eliminate, since it implies that [tex]p(x)[/tex] must have a root in [tex]\mathbb{Q}[/tex]. Since the only possible choices are [tex]\pm 1, \pm 2[/tex], it is easy to check that none of these work.

So, Case 1 is impossible.

Therefore, if [tex]p(x)[/tex] is reducible, then it must reduce into two monic polynomials of degree 2.

Case 2: [tex]p(x) = a(x)b(x)[/tex] where [tex]deg(a(x)) = deg(b(x)) = 2[/tex].

Here is where I hit a dead end. I've tried my entire bag of tricks and I have nothing to show for it. Can someone get me started?

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# Prep for Algebra Comprehensive Exam #3

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