Book edition 7th

Author(s) Kenneth H. Rosen

Pages 808 pages

ISBN 9780073383095

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Short Answer

Find the expansion of $(x + y)^{4}$ $(x + y)^{4}$
a) using combinatorial reasoning, as in Example
b) using the binomial theorem.

$x^{4} + 4 x^{3} y + 6 x^{2} y^{2} + 4 x y^{3} + y^{4}$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Formula for Binomial theorem

Binomial theorem is: $(x + y)^{n} = \sum_{j = 0}^{n} (\begin{array}{l} n \\ j \end{array}) x^{n - j} y^{j}$

Step 2: Use Distributive property and Binomial theorem

Given: $(x + y)^{4}$

(a)

Write the given statement as a product and then use the distributive property

$\begin{aligned} (x + y)^{4} = (x + y) (x + y) (x + y) (x + y) \\ (x + y)^{4} = (x^{2} + x y + x y + y^{2}) (x^{2} + x y + x y + y^{2}) \end{aligned}$

$\begin{aligned} (x + y)^{4} = (x^{2} + 2 x y + y^{2}) (x^{2} + 2 x y + y^{2}) \\ (x + y)^{4} = x^{4} + 2 x^{3} y + x^{2} y^{2} + 2 x^{3} y + 4 x^{2} y^{2} + 2 x y^{3} + x^{2} y^{2} + 2 x y^{3} + y^{4} \\ (x + y)^{4} = x^{4} + 4 x^{3} y + 6 x^{2} y^{2} + 4 x y^{3} + y^{4} \end{aligned}$

(b)

Use the binomial theorem:

$(x + y)^{4} = (\begin{array}{l} 4 \\ 0 \end{array}) x^{4} y^{0} + (\begin{array}{l} 4 \\ 1 \end{array}) x^{3} y^{1} + (\begin{array}{l} 4 \\ 2 \end{array}) x^{2} y^{2} + (\begin{array}{l} 4 \\ 3 \end{array}) x^{1} y^{3} + (\begin{array}{l} 4 \\ 4 \end{array}) x^{0} y^{4}$

$(x + y)^{4} = \frac{4!}{0! 4!} x^{4} + \frac{4!}{1! 3!} x^{3} y + \frac{4!}{2! 2!} x^{2} y^{2} + \frac{4!}{3! 1!} x y^{3} + \frac{4!}{4! 0!} y^{4}$

$\begin{aligned} (x + y)^{4} = 1 x^{4} + 4 x^{3} y + 6 x^{2} y^{2} + 4 x y^{3} + 1 y^{4} \\ (x + y)^{4} = x^{4} + 4 x^{3} y + 6 x^{2} y^{2} + 4 x y^{3} + y^{4} \end{aligned}$

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Discrete Mathematics and its Applications

Answers without the blur.

Short Answer

Find the expansion of $(x + y)^{4}$ $(x + y)^{4}$
a) using combinatorial reasoning, as in Example
b) using the binomial theorem.

Step by Step Solution

Step 1: Formula for Binomial theorem

Step 2: Use Distributive property and Binomial theorem

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Find the expansion of (x+y)4(x+y)4a) using combinatorial reasoning, as in Example b) using the binomial theorem.

Step by Step Solution

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Step 2: Use Distributive property and Binomial theorem

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Find the expansion of $(x + y)^{4}$ $(x + y)^{4}$
a) using combinatorial reasoning, as in Example
b) using the binomial theorem.