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# 1140 L16 - Rational Functions Def A rational function is a...

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Lecture 16, Part I: Section 2.5 Zeros of Polynomial Functions Linear Factorization Theorem Every polynomial function f ( x ) of degree n > 0 can be factored into n linear factors (not necessarily distinct) of the form f ( x ) = a n ( x c 1 )( x c 2 ) ( x c n ) where c 1 ; c 2 ; : : : ; c n are complex numbers. In other words, every polynomial function of degree n > 0 has exactly n (not necessarily distinct) zeros in the complex number system. ex. Find all the zeros of f ( x ) = x 4 16 in complex number system.

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Recall: Conjugate Pairs Theorem Suppose that f ( x ) is a polynomial with real coe cients. If a + bi ( b 6 = 0) is a zero of f , then the conjugate a bi is also a zero of f . Corollary: A polynomial f of odd degree with real coe cients has at least one real zero. ex. Find a polynomial of degree 4 with real coe cients that has 0, 1 and 1 + i as zeros.
Practice. 1) Find all zeros of f ( x ) = 2 x 4 + 5 x 3 + 5 x 2 + 20 x 12 given that 2 i is a zero of f . 2) Find a polynomial of degree 4 such that 1 is a zero of multiplicity 2 and 2 i is also a zero. 3) Find a polynomial of degree 4 such that 1+i is a zero and its graph is shown below.

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Lecture 16, Part II: Section 2.6

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Unformatted text preview: Rational Functions Def. A rational function is a function of the form f ( x ) = p ( x ) q ( x ) where p and q are polynomial functions and q is not the zero polynomial. Domain of a rational function: Zeros of a rational function: ex. Find the domain and zeros of the rational functions. 1) f ( x ) = x 2 x 2 ± x 2) f ( x ) = 3 x x 2 + 1 3) f ( x ) = x 2 + 4 x + 3 x 2 ± 9 Vertical Asymptotes Recall the graph of f ( x ) = 1 x Def. The line x = a is a vertical asymptote (VA) of the graph of rational function f ( x ) if NOTE: A graph DOES NOT cross its vertical asymptote(s). The graph of a rational function has a vertical asymp-tote at the zeros of the simpli±ed denominator. ex. Find the vertical asymptote of each function. 1) f ( x ) = x + 2 x 2 ± 9 2) g ( x ) = 2 x 2 + 4 3) h ( x ) = x 2 ± 2 x x 2 ± 3 x + 2 Horizontal Asymptotes Def. The line y = L is a horizontal asymptote (HA) of the graph f ( x ) = p ( x ) q ( x ) if...
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