College Algebra (10th Edition)

Published by Pearson
ISBN 10: 0321979478
ISBN 13: 978-0-32197-947-6

Chapter 5 - Section 5.1 - Polynomial Functions and Models - 5.1 Assess Your Understanding - Page 339: 61

Answer

(a) The real zeros are -4 with a multiplicity of 3 and -0.5 with a multiplicity of 2. (b) The graph crosses the x-axis at (-4,0) while the graph touches the x-axis at (-0.5,0). (c) The maximum number of turning points is 4. (d) The end behavior resembles $-2x^5$

Work Step by Step

Real zeros (or the x-intercepts) are found by making the function equal to zero and solving for x. This is made easier if the polynomial function is in the factored form $f(x)=a(x-r_1)(x-r_2)...(x-r_n)$ since we can solve all $(x-r)$'s for zero. The multiplicity is determined by the exponent that $(x-r)$ is raised by. For example, $(x-10)$ has a multiplicity of 1 while $(x+2)^3$ has a multiplicity of 3. The graph touches the x-axis when the multiplicity is even while the graph crosses the x-axis when the multiplicity is odd. The maximum number of turning points is determined by the highest degree of the function minus 1. For example, the maximum number of turning points of $x^5$ is 5-1=4. The end behavior resembles the graph of $y=a_nx^n$ where $f(x)= a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$
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