College Algebra 7th Edition

Published by Brooks Cole
ISBN 10: 1305115546
ISBN 13: 978-1-30511-554-5

Chapter 3, Polynomial and Rational Functions - Section 3.5 - Complex Zeros and the Fundamental Theorem of Algebra - 3.5 Exercises - Page 329: 35

Answer

The zeros of the function are: $\{0,\sqrt 3i,-\sqrt 3i\}$ The complete factorization of P is: $P(x)=x[(x-\sqrt 3i)(x+\sqrt 3i)]^{2}$ $x =0$ with multiplicity 1 $x =\sqrt 3i$ with multiplicity 2 $x =-\sqrt 3i$ with multiplicity 2

Work Step by Step

Factor the polynomial completely to obtain: $P(x)=x^{5}+6x^{3}+9x$ $P(x)=x(x^{4}+6x^{2}+9)$ $P(x)=x(x^{2}+3)^{2}$ $P(x)=x[(x+\sqrt 3i)(x-\sqrt 3i)]^{2}$ Equate each unique factor to zero then solve each equation to obtain: $x=0 $ $x+\sqrt 3i=0 \rightarrow x=-\sqrt 3i$ $x-\sqrt 3i=0 \rightarrow x=\sqrt 3i$ The zeros of the function are: $\{0,\sqrt 3i,-\sqrt 3i\}$ The complete factorization of P is: $P(x)=x[(x-\sqrt 3i)(x-(-\sqrt 3i))]^{2}$ $P(x)=x[(x-\sqrt 3i)(x+\sqrt 3i)]^{2}$ $x =0$ with multiplicity 1 $x =\sqrt 3i$ with multiplicity 2 $x =-\sqrt 3i$ with multiplicity 2
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