Precalculus (6th Edition)

Published by Pearson
ISBN 10: 013421742X
ISBN 13: 978-0-13421-742-0

Chapter 3 - Polynomial and Rational Functions - 3.3 Zeros of Polynomial Functions - 3.3 Exercises - Page 338: 113

Answer

$\{3,3,1\pm i\sqrt 7\}$

Work Step by Step

Step 1. Given $f(x)=x^4-8x^3+29x^2-66x+72$, we can list possible rational zeros as $\pm1,\pm2,\pm3,\pm4,...$ Step 2. There are 4 sign changes in $f(x)$ indicating that there could be 4, 2 or 0 positive real zeros. Step 3. $f(-x)=x^4+8x^3+29x^2+66x+72$, there is 0 sign changes in $f(-x)$ indicating that there will be 0 negative real zero. Step 4. Use synthetic division to find two real zeros as shown in the figure. Step 5. The resulting quotient gives $x^2-2x+8=0$ which gives $x=\frac{2\pm\sqrt {2^2-4(8)}}{2}=1\pm i\sqrt 7$ Step 6. The zeros are $\{3,3,1\pm i\sqrt 7\}$
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