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ss_3_3

# ss_3_3 - Section 3.3 Polynomial Functions Polynomial of...

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Section 3.3 Polynomial Functions Polynomial of degree n where n is a non-negative integer and a n 6 = 0: f ( x ) = a n x n + a n - 1 x n - 1 + · · · + a 1 x + a 0 Example. Polynomial function: f ( x ) = - 2 x 3 + 4 x Example. Polynomial function: f ( x ) = 4 Example. NOT a polynomial function: f ( x ) = x - 3 + x Example. NOT a polynomial function: f ( x ) = x Graphs of polynomial functions are continuous (no breaks) smooth (no sharp corners) Graph of y = ( x + 1)( x )( x - 1)( x - 2): –2 –1 0 1 2 –2 –1 1 2 Finding a Polynomial with given Zeros: Example. Find a polynomial of degree 3, with zeros at 3, -2, 1. Solution. f ( x ) = a ( x - 3)( x - ( - 2))( x - 1) = a ( x - 3)( x + 2)( x - 1), with a 6 = 0 arbitrary. The graph of f is shown below with a = 1. –12 –6 0 6 12 –3 –2 –1 1 2 3 4 y = ( x - 3)( x + 2)( x - 1) Example. Find a polynomial with zeros of 3, -2, 1 having degree 4 and with 3 a zero of multiplicity two. Solution. f ( x ) = a ( x - 3) 2 ( x - ( - 2))( x - 1) = a ( x - 3) 2 ( x + 2)( x - 1), with a 6 = 0 arbitrary. 1

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–30 –20 –10 10 –3 –1 1 3 y = ( x - 3) 2 ( x + 2)( x - 1) The role of multiplicity in graphing polynomials: The graphs of f ( x ) = a ( x - 1)( x - 2) n ( x -
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ss_3_3 - Section 3.3 Polynomial Functions Polynomial of...

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