Precalculus (10th Edition)

Published by Pearson
ISBN 10: 0-32197-907-9
ISBN 13: 978-0-32197-907-0

Chapter 12 - Sequences; Induction; the Binomial Theorem - 12.5 The Binomial Theorem - 12.5 Assess Your Understanding - Page 836: 40

Answer

$-84$

Work Step by Step

We are given the expression: $\left(x-\dfrac{1}{x^2}\right)^9$ The term $T_{k+1}$ of the expansion of $(a+b)^n$ is: $T_{k+1}=\binom{n}{k}a^{n-k}b^r$ We have: $T_{k+1}=\binom{9}{k}x^{9-k}(-x^{-2})^k=(-1)^k\binom{9}{k}x^{9-k-2k}=(-1)^k\binom{9}{k}x^{9-3k}$ Determine $k$ so that the term contains $x^0$: $9-3k=0$ $9=3k$ $k=3$ Determine the coefficient of $x^0$: $(-1)^3\binom{9}{3}=-\dfrac{9!}{3!6!}=-84$
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