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 KŒxŁ . 1.8.12. Make a reasonable deﬁnition 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 ²²² 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 smallest degree and that I D fKŒxŁ . 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 ﬁnite collection of nonzero polynomials with real coefﬁcients....
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 Fall '08
 EVERAGE
 Algebra, Polynomials, Division, Remainder, Prime number, Greatest common divisor, 1.8.9.

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