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: 36

Answer

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

Work Step by Step

Factor the polynomial completely to obtain: $P(x)=x^{6}+16x^{3}+64$ $P(x)=(x^{3}+8)^{2}$ $P(x)=[(x+2)(x^{2}-2x+4)]^{2}$ $P(x)=[(x+2)(x-(1+\sqrt 3i))(x-(1-\sqrt 3i))]^{2}$ Equate each unique factor to zero then solve each equation to obtain: $x+2=0 \rightarrow x=-2$ $x-(1+\sqrt 3i)=0 \rightarrow x=1+\sqrt 3i$ $x-(1-\sqrt 3i)=0 \rightarrow x=1-\sqrt 3i$ The zeros of the function are: $\{-2,1+\sqrt 3i,1-\sqrt 3i\}$ The complete factorization of P is: $P(x)=[(x+2)(x-(1+\sqrt 3i))(x-(1-\sqrt 3i))]^{2}$ $x =-2$ with multiplicity 2 $x =1+\sqrt 3i$ with multiplicity 2 $x =1-\sqrt 3i$ with multiplicity 2
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