College Algebra Exam Review 45

College Algebra Exam Review 45 - r;s such that rf C sg D g...

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1.8. POLYNOMIALS 55 1.8.5. Let h be a non-zero element of I.f;g/ of least degree. Show that h is a greatest common divisor of f and g . Hint: Apply division with remainder. 1.8.6. Show that two polynomials f;g 2 KŒxŁ are relatively prime if, and only if, 1 2 I.f;g/ . 1.8.7. Show that if p 2 KŒxŁ is irreducible and f 2 KŒxŁ , then either p divides f , or p and f are relatively prime. 1.8.8. Let p 2 KŒxŁ be irreducible. Prove the following statements. (a) If p divides a product fg of elements of KŒxŁ , then p divides f or p divides g . (b) If p divides a product f 1 f 2 :::f r of several elements of KŒxŁ , then p divides one of the f i . Hint: Mimic the arguments of Proposition 1.6.16 and Corollary 1.6.17 . 1.8.9. Prove Theorem 1.8.21 . (Mimic the proof of Theorem 1.6.18 . ) 1.8.10. For each of the following pairs of polynomials f;g , find the great- est common divisor and find polynomials r;s such that rf C sg D g : c : d :.f;g/ . (a) x 3 ± 3x C 3 , x 2 ± 4 (b) ± 4 C 6x ± 4x 2 C x 3 , x 2 ± 4 1.8.11. Write a computer program to compute the greatest common divisor of two polynomials f;g with real coefficients. Make your program find polynomials
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Unformatted text preview: r;s such that rf C sg D g : c : d :.f;g/ . The next three exercises explore the idea of the greatest common divisor of several nonzero polynomials, f 1 ;f 2 ;:::;f k 2 Kx . 1.8.12. Make a reasonable denition of g : c : d .f 1 ;f 2 ;:::;f k / , and show that g : c : d .f 1 ;f 2 ;:::;f k / D g : c : d :.f 1 ; g : c : d .f 2 ;:::;f k /// . 1.8.13. (a) Let I D I.f 1 ;f 2 ;:::;f k / D f m 1 f 1 C m 2 f 2 C C m k f k W m 1 ;:::;m k 2 Z g : Show that I has all the properties of I.f;g/ listed in Proposition 1.8.15 . (b) Show that f D g : c : d .f 1 ;f 2 ;:::;f k / is an element of I of small-est degree and that I D fKx . 1.8.14. (a) Develop an algorithm to compute g : c : d .f 1 ;f 2 ;:::;f k / . (b) Develop a computer program to compute the greatest common divisor of any nite collection of nonzero polynomials with real coefcients....
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