Question

factor the sum of two squares.
x² + y²

x² + y² =
(type an exact answer, using radicals and i as needed.)

Explanation:

Step1: Recall complex - number factoring

In the complex - number system, we know that \(a^{2}-b^{2}=(a + b)(a - b)\). We can rewrite \(x^{2}+y^{2}\) as \(x^{2}-(-y^{2})\).
Since \(-y^{2}=(iy)^{2}\), then \(x^{2}+y^{2}=x^{2}-(iy)^{2}\).

Step2: Apply the difference - of - squares formula

Using the formula \(a^{2}-b^{2}=(a + b)(a - b)\) with \(a = x\) and \(b=iy\), we get \(x^{2}-(iy)^{2}=(x + iy)(x-iy)\).

Answer:

\((x + iy)(x - iy)\)