College Algebra (10th Edition)

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

Chapter 5 - Section 5.6 - Complex Zeros; Fundamental Theorem of Algebra - 5.6 Assess Your Understanding: 23


$4$ and $-2i$

Work Step by Step

RECALL: If $a+bi$ is a zero of a polynomial function with real number coefficients, then its conjugate $a-bi$ is also a zero of the function. Thus, one of the missing zeros of the given function is the conjugate of $2i$, which is ${\bf -2i}$. Thus, two of the three zeros of the polynomial function with real coefficients are: $2i$ and $-2i$ The third zero is a real number and can be found by factoring the polynomial: $P(x) = x^3-4x^2+4x-16 \\P(x) = (x^3-4x^2)+(4x-16) \\P(x) = x^2(x-4)+4(x-4) \\P(x) = (x-4)(x^2+4)$ Equating each factor to zero gives: \begin{array}{ccc} &x-4=0 &\text{or} &x^2+4=0 \\&x=4 &\text{or} &x^2=-4 \end{array} Thus, the real zero of the function is $4$. Therefore, the missing zeros are: $4$ and $-2i$
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