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

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.
Ask a homework question - tutors are online