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Chapter 2: Problem 80

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. In the complex number system, \(x^{2}+y^{2}\) (the sum of two squares) can be factored as \((x+y i)(x-y i)\)

Short Answer

Expert verified
The statement 'In the complex number system, \(x^{2}+y^{2}\) (the sum of two squares) can be factored as \((x+y i)(x-y i)\)' is false. The correct statement should be 'In the complex number system, \(x^{2}+y^{2}\) (the sum of two squares) is equivalent to \((x+y i)(x-y i) = x^{2} + y^{2}\)'.

Step by step solution

01

Check the Statement

The first step is to verify the given statement. By multiplying \((x+y i)(x-y i)\), we get \(x^{2} -(y i)^{2} = x^{2} + y^{2}\). This is because \(i^{2} = -1\) in complex number system. Thus, the statement is false as it does not yield the same as the original statement \(x^{2} + y^{2}\). The correct form should be \(x^{2} -(y i)^{2} = x^{2} + y^{2}\).
02

Correct the Statement

On the basis of the above verification, correct the statement to match the original. Take the given statement and replace \((x+y i)(x-y i)\) with \((x+y i)(x-y i) = x^{2} + y^{2}\) instead of \( x^{2}+y^{2} = (x+y i)(x-y i)\), which was the initial version.

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