College Algebra Exam Review 46

College Algebra Exam Review 46 - 56 1 ALGEBRAIC THEMES...

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Unformatted text preview: 56 1. ALGEBRAIC THEMES 1.8.15. (a) Suppose that p.x/ D a n x n C C a 1 x C a 2 ZOExŁ . Suppose that r=s 2 Q is a root of p , where r and s are relatively prime integers. Show that s divides a n and r divides a . Hint: Start with the equation p.r=s/ D , multiply by s n , and note, for example, that all the terms except a n r n are divisible by s . (b) Conclude that any rational root of a monic polynomial in Z OExŁ is an integer. (c) Conclude that x 2 2 has no rational root, and therefore p 2 is irrational. 1.8.16. (a) Show that a quadratic or cubic polynomial f.x/ in KOExŁ is irre- ducible if, and only if, f.x/ has no root in K . (b) A monic quadratic or cubic polynomial f.x/ 2 Z OExŁ is irre- ducible if, and only if, it has no integer root. (c) x 3 3x C 1 is irreducible in Q OExŁ . 1.9. Counting Counting is a fundamental and pervasive technique in algebra. In this sec- tion we will discuss two basic counting tools, the binomial coefficients and the method of inclusion-exclusion. These tools will be used to estab-and the method of inclusion-exclusion....
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