Problem 74 - College Algebra Essentials

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 given statement that in the complex number system, the sum of two squares \(x^{2}+y^{2}\) can be factored as \((x+y i)(x-y i)\) is true.

Step 1: Analyze the given statement

In the given statement, it is stated that the sum of squares of two real numbers, \(x^{2}+y^{2}\), can be factored as \((x+y i)(x-y i)\). To verify if the statement is true or false, expand the expression \((x+y i)(x-y i)\) using the distributive property.

Step 2: Expand the expression

Expand the expression \((x+y i)(x-y i)\) using the standard multiplication rule for binomials which are \((a+b)(a-b) = a^{2} - b^{2}\). Here \(a=x\) and \(b=yi\). So we get \(x^{2} - (yi)^{2}\). Further simplify to \(x^{2} - (-y^{2}) = x^{2}+ y^{2}\).

Step 3: Compare with the original expression

The obtained expression is \(x^{2}+ y^{2}\). Comparing this to the original sum of squares, it matches exactly.

Step 4: State the conclusion

Since the original expression and the expanded expression are the same, the given statement is true.