Exploring Polynomials: Zeros, Factors, and Graphs
Understanding polynomials, linear factors, and zeros. Learn how to write and graph polynomial functions, find roots and x-intercepts, apply the Factor Theorem, and plot graphs using zeros and end behaviors.
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5-2 Polynomials, Linear Factors, & Zeros Today s Objective: I can write and graph a polynomial function
Roots, Zeros & x-intercepts ? ? = ????+ ?? ??? ?+ + ??? + ?? Factor Theorem ? ? is a linear factor of the polynomial ?(?) if and only if b is a zero of the polynomial function ?(?) Find the zeros ? = ?3+ 2?2 24? Write the polynomial given the zeros: 2, 2,3 ? = ? 1 ? + 2 (? 3) ? = (? )(? )(? ) 2 +2 3 ? = ?(?2+ 2? 24) 1, 2,3 ? = (?2 4)(? 3) ? = ?3 3?2 4? + 12 ? = ? ? 4 (? + 6) ? = ? ? + 5 ? 7 0, 5,7 0,4, 6
Graphing with zeros ?(?) = ?(? 4)(? + 3) 1. Find and plot the zeros 2. Sketch end behavior 3. Pick easy midpoints between zeros to estimate turning point Zeros: 0,4, 3 ?( 2) = 2( 2 4)( 2 + 3) = 12 ?(2) =2(2 4)(2 + 3) = 20
Zeros with Multiplicity ?(?) = (? + 2)2(? 2)(? 3) 1. Find and plot the zeros 2. Sketch end behavior 3. Pick easy midpoints between zeros to estimate turning point ?(?) = (? + 2)(? + 2)(? 2)(? 3) Even multiplicity turns graph at zero Odd multiplicity pauses graph zero Zeros: 2, 2,2,3 (0 + 2)2(0 2)(0 3) ?(0) = = 24 2.5 + 22(2.5 2)(2.5 3) ?(2.5) = p. 293:9-33 mult. of 3 = 5
