College Algebra Exam Review 42

College Algebra Exam Review 42 - such that d.x/ D s.x/ f.x/...

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52 1. ALGEBRAIC THEMES steps: g D q 1 f C f 1 f D q 2 f 1 C f 2 : : : f k ± 2 D q k f k ± 1 C f k : : : f r ± 1 D q r C 1 f r : Here deg f > deg .f 1 / > deg .f 2 / > ::: . By the argument of Proposition 1.6.10 , the final nonzero remainder f r is an element of I.f;g/ and is a greatest common divisor of f and g . n Example 1.8.17. Compute the (monic) greatest common divisor of f.x/ D ± 4 C 9x ± 3x 2 ± 6x 3 C 6x 4 ± 3x 5 C x 6 and g.x/ D 3 ± 6x C 4x 2 ± 2x 3 C x 4 : Repeated division with remainder gives . ± 4 C 9x ± 3x 2 ± 6x 3 C 6x 4 ± 3x 5 C x 6 / D . ± x C x 2 /.3 ± 6x C 4x 2 ± 2x 3 C x 4 / C . ± 4 C 12x ± 12x 2 C 4x 3 /; .3 ± 6x C 4x 2 ± 2x 3 C x 4 / D . x 4 C 1 4 /. ± 4 C 12x ± 12x 2 C 4x 3 / C .4 ± 8x C 4x 2 /; . ± 4 C 12x ± 12x 2 C 4x 3 / D . ± 1 C x /.4 ± 8x C 4x 2 / C 0: Thus a (non-monic) greatest common divisor is d.x/ D 4 ± 8x C 4x 2 : We can find the coefficients s.x/;t.x/
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Unformatted text preview: such that d.x/ D s.x/ f.x/ C t.x/ g.x/ as follows: The sequence of quotients produced in the algorithm is q 1 D x C x 2 , q 2 D x 4 C 1 4 , and q 3 D 1 C x . The q k determine matrices Q k D 1 1 q k . The coefcients s.x/;t.x/ comprise the rst column of the product Q D Q 1 Q 2 Q 3 . (Compare the proof of Proposition 1.6.10 and Example 1.6.11 .) The result is s.x/ D x 4 1 4 t.x/ D x 3 4 x 4 C 1:...
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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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