4.5%20Irreducibility%20in%20Q%5bx%5d

# 4.5%20Irreducibility%20in%20Q%5bx%5d - Math 302 Homework...

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Math 302 Modern Algebra Spring 2009 Homework Solutions 4.5 IRREDUCIBILITY IN Q [ x ] 1. Use the Rational Root Test to write each polynomial as a product of irreducible polynomials in Q [ x ]. (e) 2 x 4 + 7 x 3 + 5 x 2 + 7 x + 3 By the Rational Root Test, if r/s is a rational root, then r | 3 and s | 2, so r = ± 1 , ± 3 and s = ± 1 , ± 2 . Therefore the possible rational roots are r/s = ± 1 2 , ± 1 , ± 3 2 , ± 3 . Testing the roots one by one, we Fnd that 1 2 and 3 are roots and the polynomial can be factored as 2 x 4 + 7 x 3 + 5 x 2 + 7 x + 3 = 2( x + 1 2 )( x + 3)( x 2 + 1) . Since x + 1 2 and x + 3 are linear, then both are irreducible in Q [ x ]. Since x 2 + 1 has no rational roots, then it’s irreducible in Q [ x ] by Corollary 4.18. (f) 6 x 4 31 x 3 + 25 x 2 + 33 x + 7 By the Rational Root Test, if r/s is a rational root, then r | 7 and s | 6, so r = ± 1 , ± 7 and s = ± 1 , ± 2 , ± 3 , ± 6 . Therefore the possible rational roots are
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