Precalculus: Mathematics for Calculus, 7th Edition

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

Chapter 12 - Section 12.6 - The Binomial Theorem - 12.6 Exercises - Page 886: 43

Answer

$(x+y)^{4}$

Work Step by Step

$(a+b)^{n}=\left(\begin{array}{l} n\\ 0 \end{array}\right)a^{n}+\left(\begin{array}{l} n\\ 1 \end{array}\right)a^{n-1}b+\left(\begin{array}{l} n\\ 2 \end{array}\right)a^{n-2}b^{2}+\cdots+\left(\begin{array}{l} n\\ n \end{array}\right)b^{n}$ ------------------- The coefficients are 1, 4, 6, 4, 1 $\left(\begin{array}{l} 4\\ 0 \end{array}\right)=1=\left(\begin{array}{l} 4\\ 4 \end{array}\right)$ $\left(\begin{array}{l} 4\\ 1 \end{array}\right)=4=\left(\begin{array}{l} 4\\ 3 \end{array}\right)$ $\displaystyle \left(\begin{array}{l} 4\\ 2 \end{array}\right)=\frac{4\times 3}{1\times 2}=6$ The exponents of x decrease from 4 to zero The exponents of y increase from 0 to 4. $x^{4}+4x^{3}y+6x^{2}y^{2}+4xy^{3}+y^{4}=(x+y)^{4}$
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