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Unformatted text preview: Factor Theorem and Intro to Cardinality Original Notes adopted from November 27, 2001 (Week 12) c P. Rosenthal , MAT246Y1, University of Toronto, Department of Mathematics If a and b are irrational, can a b be rational? Yes. Eg . ( 3) 2 (( 3) 2 ) 2 = ( 3) 2 = 3 . ( 3) 2 is either rational or irrational. If rational, its an example. If irrational, [( 3) 2 )] 2 is an example. Therefore there exists a,b irrational such that a b rational. Theorem. ( Factor Theorem ) If p is a polynomial, r C , then p ( r ) = 0 iff there exists polynomial q such that p ( z ) = ( z- r ) q ( z ). Corollary. If p is a polynomial of degree n over C , then there exists r 1 ,r 2 , ,r n ,c C such that p ( z ) = c ( z- r 1 )( z- r 2 ) ( z- r n ). Proof By Fundamental Theorem of Algebra, if p not constant, p has a root. By factor theorem, p has a factor of form ( z- n ). Keep applying this procedure. Corollary. If p polynomial of degree n over C , then there exists r 1 , r 2 ......r n C with r 1 6 = r j for i 6 = j and natural numbers k 1 ,k 2 , ,k n , and c C such that p ( z ) = c ( z- r 1 ) k 1 ( z- r 2 ) k 2 ........ ( z- r m ) k m ....
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