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: 21

Answer

$\color{blue}{P(x)=x^4-6x^3+10x^2-6x+9}$

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, the missing zero of the given function is the conjugates of $-i$, which is ${\bf i}$. The zeros of the polynomial function with real coefficients are: $3$ (multiplicity 2) $\\-i \\i$ This means that the function is: $P(x) = a(x+i) (x-i)(x-3)(x-3) \\P(x) = a(x^2-i^2)(x^2-6x+9) \\P(x) =ax^2-(-1)](x^2-6x+9)[ \\P(x) = a(x^2+1)(x^2-6x+9) \\P(x) = a(x^4-6x^3+9x^2+x^2-6x+9) \\P(x) = a(x^4-6x^3+10x^2-6x+9)$ Setting the leading coefficient $a=1$ gives: $\color{blue}{P(x)=x^4-6x^3+10x^2-6x+9}$
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