Problem 74 - College Algebra

Question

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 \((x+y i)(x-y i)\)

Step-by-Step Solution

Answer

The statement is false. The correct statement is: In the complex number system, \(x^{2}+y^{2}\) can be factored as \((x+ yi)(x- yi)\), where y is a real number and i is the imaginary unit.

Step 1: Understanding complex numbers

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit, which satisfies the equation \(i^{2} = -1\). iota (i) can be thought of as the square root of -1. Therefore, when dealing with complex numbers, \(x^{2}+y^{2}\) doesn't factorize into \((x+y i)(x-y i)\) as there is no imaginary part in \(x^{2}+y^{2}\).

Step 2: Comparison with true identity

However, considering the formula for multiplication of two complex numbers, if you multiply \((x+y i)\) by \((x-y i)\), which are essentially complex conjugates of each other, the product would be \(x^2 + y^2\). What this means is that the factors would actually be \(x + yi\) and \(x - yi\) multiplied, where y is also considered a real number and i is an imaginary unit, not \(x + y\) and \(x - y\). The given statement misinterpreted the i as part of the number instead of as an imaginary unit.

Step 3: Correcting the Statement

Rewrite the expression in the corrected form. This makes the correct statement: In the complex number system, \(x^{2}+y^{2}\) can be factored as \((x+yi)(x-yi)\), where y is a real number and i is the imaginary unit.