Q 26. In Problem, prove each statement... [FREE SOLUTION]

91影视

Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 12: Q 26. (page 830)

In Problem, prove each statement.
$a + b$ is a factor of $a^{2 n + 1} + b^{2 n + 1}$

Short Answer

Expert verified

The given statement is shown.

Step by step solution

Step 1. Given information

The given expression is $a + b$ .

Step 2. Prove the given statement.

First we consider the statement for $n = 1$ .

$\begin{array}{rcl} a^{2.1 + 1} + b^{2.1 + 1} & = & a^{3} + b^{3} \\ = & (a + b) (a^{2} + b^{2} - a b) \end{array}$

which is divisible by $a + b$

Hence, condition holds true.

Now we assume the statement is true for some $k$

Now we consider,

$\begin{array}{rcl} a^{2 (k + 1) + 1} + b^{2 (k + 1) + 1} & = & a^{2 k + 3} + b^{2 k + 3} \\ = & a^{2 k + 1} 路 a^{2} + b^{2 k + 1} 路 b^{2} \\ = & (a^{2 k + 1} + b^{2 k + 1} - b^{2 k + 1}) 路 a^{2} + b^{2 k + 1} b^{2} \\ = & (a^{2 k + 1} + b^{2 k + 1}) a^{2} - a^{2} b^{2 k + 1} + b^{2 k + 1} b^{2} \\ = & (a^{2 k + 1} + b^{2 k + 1}) a^{2} + b^{2 k + 1} (b^{2} - a^{2}) \end{array}$

Since $(a^{2 k + 1} + b^{2 k + 1})$ is divisible by $a + b$ and $b^{2 k + 1} (b^{2} - a^{2})$ is also divisible by $a + b$ , we say that condition also holds true.

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91影视

In Problem, prove each statement.
$a + b$ is a factor of $a^{2 n + 1} + b^{2 n + 1}$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Prove the given statement.

One App. One Place for Learning.

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91影视

In Problem, prove each statement.a+bis a factor ofa2n+1+b2n+1

Short Answer

Step by step solution

Step 1. Given information

Step 2. Prove the given statement.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Statistics

Decision Maths

Mechanics Maths

Applied Mathematics

Pure Maths

Study anywhere. Anytime. Across all devices.

In Problem, prove each statement.
$a + b$ is a factor of $a^{2 n + 1} + b^{2 n + 1}$