mac1105_lecture15_1

# mac1105_lecture15_1 - L15 Polynomial Functions Polynomial...

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155 L15 Polynomial Functions; Polynomial Inequalities A polynomial function is a function of the form 1 11 0 () . . . nn fx ax a x ax a =+ + + + where 0 , ,..., , aa a a are real numbers and 0 n is an integer. Degree 0: 0 fx a = , 0 0 a Degree 1: 10 fx ax a , 1 0 a Degree 2: 2 21 0 f xa x a + , 2 0 a Features of the Polynomial Function : 1. The domain is the whole real line. 2. The graph has no holes, gaps, or jumps – a polynomial function is continuous . 3. The graph has no sharp corners or cusps – a polynomial function is smooth . Power Function A power function of degree n is a function of the form n fx x = where 0 n > is an integer.

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156 Properties of the Power Functions : 1) n yx = , where n is even : Symmetry: End Behavior: 2) n = , where n is odd : Symmetry: End Behavior: Note : The greater the value of n , the steeper graph of n = when 1 x < − or 1 x > ; and the flatter graph of n = when 11 x −< < .
157 Graphing Polynomials A number c is called a zero of a polynomial ( ) fx if () 0 fc = . x c = is a zero of a polynomial ( ) if and only if () x c is a factor of f . The number of times the factor ( ) x c occurs in the polynomial is called the multiplicity of the zero c . Example : Find all zeros and their multiplicities. (Give the multiplicity in parentheses next to a zero.) 23 2 2 ( 4 )( 2 ) ( 1 ) x x x x =− + + The Fundamental Theorem of Algebra : Every polynomial of degree 1 or more has at least one complex zero.

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## This note was uploaded on 11/07/2011 for the course MAC 1105 taught by Professor Picklesimer during the Fall '10 term at University of Florida.

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mac1105_lecture15_1 - L15 Polynomial Functions Polynomial...

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