College Algebra Exam Review 305

College Algebra Exam Review 305 - polynomial of degree n...

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6.8. IRREDUCIBILITY CRITERIA 315 where p is a prime, we have f.x C 1/ D p ± 1 X s D 0 ± p s C 1 ² x s : This is irreducible by Eisenstein’s criterion, so f is irreducible as well. You are asked to provide the details for this example in Exercise 6.8.2 . There is a simple criterion for a polynomial in Z ŒxŁ to have (or not to have) a linear factor, the so–called rational root test . Proposition 6.8.7. (Rational root test) Let f.x/ D a n x n C a n ± 1 x n ± 1 C ²²² a 1 x C a 0 2 Z ŒxŁ . If r=s is a rational root of f , where r and s are relatively prime, then s divides a n and r divides a 0 . Proof. Exercise 6.6.3 . n A quadratic or cubic polynomial is irreducible if, and only if, it has no linear factors, so the rational root test is a definitive test for irreducibility for such polynomials. The rational root test can sometimes be used as an adjunct to prove irreducibility of higher degree polynomials: If an integer
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Unformatted text preview: polynomial of degree n has no rational root, but for some prime p its re-duction mod p has irreducible factors of degrees 1 and n ± 1 , then then f is irreducible (Exercise 6.8.1 ). Exercises 6.8 6.8.1. Show that if a polynomial f.x/ 2 Z ŒxŁ of degree n has no rational root, but for some prime p its reduction mod p has irreducible factors of degrees 1 and n ± 1 , then then f is irreducible. 6.8.2. Provide the details for Example 6.8.6 . 6.8.3. Show that for each natural number n .x ± 1/.x ± 2/.x ± 3/ ²²² .x ± n/ ± 1 is irreducible over the rationals. 6.8.4. Show that for each natural number n ¤ 4 .x ± 1/.x ± 2/.x ± 3/ ²²² .x ± n/ C 1 is irreducible over the rationals....
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

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