College Algebra (10th Edition)

Published by Pearson
ISBN 10: 0321979478
ISBN 13: 978-0-32197-947-6

Chapter 9 - Section 9.5 - The Binomial Theorem - 9.5 Assess Your Understanding: 19

Answer

$(x-2)^6=(x+(-2))^6=\binom{6}{0}x^6(-2)^0+\binom{6}{1}x^{5}(-2)^1+\binom{6}{2}x^{4}(-2)^2+\binom{6}{3}x^{3}(-2)^3+\binom{6}{4}x^{2}(-2)^{4}+\binom{6}{5}x^1(-2)^5+\binom{6}{6}x^0(-2)^6=x^6-12x^5+60x^4-160x^3+240x^2-192x+64$

Work Step by Step

The binomial theorem expands an algebraic expression in the form of: $(x+y)^n=\binom{n}{0}x^ny^0+\binom{n}{1}x^{n-1}y^1+\binom{n}{2}x^{n-2}y^2+...+\binom{n}{i}x^{n-i}y^i+...+\binom{n}{n-1}x^{1}y^{n-1}+\binom{n}{n}x^0y^n=\sum_{i=0}^{n}\binom{n}{i}x^{n-i}y^i$ Here, $y=-2; n=6$ $(x-2)^6=(x+(-2))^6=\binom{6}{0}x^6(-2)^0+\binom{6}{1}x^{5}(-2)^1+\binom{6}{2}x^{4}(-2)^2+\binom{6}{3}x^{3}(-2)^3+\binom{6}{4}x^{2}(-2)^{4}+\binom{6}{5}x^1(-2)^5+\binom{6}{6}x^0(-2)^6=x^6-12x^5+60x^4-160x^3+240x^2-192x+64$
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