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Lecture13Complex Zeros - conjugate Corollary A complex...

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Complex Zeros; Fundamental Theorem of Algebra
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A complex polynomial function f degree n is a complex function of the form A complex number r is called a (complex) zero of a complex function f if f ( r ) = 0. Where are complex numbers. n a a a ,... , 1 0
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Fundamental Theorem of Algebra Every complex polynomial function f ( x ) of degree n > 1 has at least one complex zero.
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Fundamental Theorem of Algebra Every complex polynomial function f ( x ) of degree n > 1 can be factored into n linear factors (not necessarily distinct) of the form
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Find the zeros of Use the zeros to factor f According to the quadratic formula 5 4 ) ( 2 + - = x x x f i i x ± = ± = - ± = - ± = - ± = 2 2 2 4 2 1 2 4 2 4 4 ) 1 ( 2 ) 5 )( 1 ( 4 ) 4 ( 4 2 ) 2 )( 2 ( )) 2 ( ))( 2 ( ( ) ( i x i x i x i x x f + - - - = - - + - =
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Conjugate Pairs Theorem Let f ( x ) be a complex polynomial whose coefficients are real numbers . If r = a + bi is a zero of f , then the complex
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Unformatted text preview: conjugate Corollary A complex polynomial f of odd degree with real coefficients has at least one real zero. is also a zero of f . Find a polynomial f of degree 4 whose coefficients are real numbers and that has zeros 1, 2, and 2+ i . f(x) Given where all the coefficients are real. a) What is the maximum number of real zeros that f can have? b) What is the minimum number of zeros that f can have? c) What is the maximum number of complex (but not real) zeros of f? 4 4 5 5 ) ( a x a x a x f + + = Find the complex zeroes of the polynomial function There are 4 complex zeros. From Rational Zero Theorem find potential rational zeros Zeros are -2, -1/2, 2 +5 i , 2 -5 i ....
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