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: 24

Answer

The remaining zeros are: $-3$ and $5i$

Work Step by Step

The polynomial function's degree is three so it has three zeros. 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 $-5i$, which is ${\bf 5i}$. Thus, two of the three zeros of the polynomial function with real coefficients are: $-5i$ and $5i$ The third zero is a real number and can be found by factoring the polynomial: $P(x) = x^3+3x^2+25x+75 \\P(x) = (x^3+3x^2)+(25x+75) \\P(x) = x^2(x+3)+25(x+3) \\P(x) = (x+3)(x^2+25)$ Equating each factor to zero gives: \begin{array}{ccc} &x+3=0 &\text{or} &x^2+25=0 \\&x=-3 &\text{or} &x^2=-25 \end{array} Thus, the real zero of the function is $-3$. Therefore, the missing zeros are: $-3$ and $5i$
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