1. find the content of 1.6x^3+4x^2+1.6x+2 and the associated primitive polynomial.
2. prove that x^n+6x+3 is irreducible over Q for any n>0.
please steps by steps
1.
1.6=8/5 which is a irreducible rational fraction.
So 1.6x^3+4x^2+1.6x+2=8/5x^3+20/5x^2+8/5x+10
=(8x^3+20x^2+8x+10)/5
=2/5(4x^3+10x^2+4x+5)
So its content is 2/5 and the associated primitive polynomial is 4x^3+10x^2+4x+5.
2.
By Eisenstein's criterion, (http://en.wikipedia.org/wiki/Eisenstein%27s_criter...
we take p=3, then we get the result directly.
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Verified answer
1.
1.6=8/5 which is a irreducible rational fraction.
So 1.6x^3+4x^2+1.6x+2=8/5x^3+20/5x^2+8/5x+10
=(8x^3+20x^2+8x+10)/5
=2/5(4x^3+10x^2+4x+5)
So its content is 2/5 and the associated primitive polynomial is 4x^3+10x^2+4x+5.
2.
By Eisenstein's criterion, (http://en.wikipedia.org/wiki/Eisenstein%27s_criter...
we take p=3, then we get the result directly.