Precalculus: Mathematics for Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 1305071751
ISBN 13: 978-1-30507-175-9

Chapter 3 - Section 3.4 - Real Zeros of Polynomials - 3.4 Exercises - Page 284: 73

Answer

1. Upper bound $b=1$, see proof below. 2. Lower bound $a=-2$, see proof below.

Work Step by Step

1. Upper bound $b=1$, we use synthetic division to divide the polynomial by $x-1$ as shown in the figure. As can be seen, all the numbers $1,3,6,11,10$ in the line containing the quotient and remainder of this division are positive. Based on the Upper and Lower Bound Theorem (page 279 in the book), we conclude that $b=1$ is an upper bound of the function $P(x)$. 2. Lower bound $a=-2$, similarly, we used synthetic division to divide $P(x)$ by $x+2$ as shown in the figure, and it can be seen that the numbers $1,0,3,-1,1$ (zero is considered negative here as it can be regarded as both positive and negative) on the quotient and remainder line have alternating signs. Based on the Upper and Lower Bound Theorem (page 279 in the book), we conclude that $a=-2$ is a lower bound of the function $P(x)$.
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