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College Algebra Exam Review 43

College Algebra Exam Review 43 - pearing are unique up to...

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1.8. POLYNOMIALS 53 The monic greatest common divisor of f.x/ and g.x/ is d 1 .x/ D .1=4/d.x/ . We have d 1 .x/ D .1=4/s.x/f.x/ C .1=4/t.x/g.x/: Definition 1.8.18. Two polynomials f;g 2 KŒxŁ are relatively prime if g : c : d :.f;g/ D 1 . Proposition 1.8.19. Two polynomials f;g 2 KŒxŁ are relatively prime if, and only if, 1 2 I.f;g/ . Proof. Exercise 1.8.5 . n Proposition 1.8.20. (a) Let p be an irreducible polynomial in KŒxŁ and f;g 2 KŒxŁ nonzero polynomials. If p divides the product fg , then p divides f or p divides g . (b) Suppose that an irreducible polynomial p 2 KŒxŁ divides a prod- uct f 1 f 2 ±±± f s of nonzero polynomials. Then p divides one of the factors. Proof. Exercise 1.8.8 . n Theorem 1.8.21. The factorization of a polynomial in KŒxŁ into irre- ducible factors is essentially unique. That is, the irreducible factors ap-
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Unformatted text preview: pearing are unique up to multiplication by nonzero elements in K . Proof. Exercise 1.8.9 . n This completes our treatment of unique factorization of polynomials. Before we leave the topic, let us notice that you haven’t yet learned any general methods for recognizing irreducible polynomials, or for carrying out the factorization of a polynomial by irreducible polynomials. In the integers, you could, at least in principle, test whether a number n is prime, and find its prime factors if it is composite, by searching for divisors among the natural numbers ² p n . For an infinite field such as Q , we cannot...
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