Factoring the expression x² + y² can be a bit tricky because it doesn’t factor in the same straightforward way as other algebraic expressions. Unlike expressions that can be easily factored into real-number products, x² + y² does not have real factors when considered over the set of real numbers.
However, we can still express it as a factorable form using complex numbers. The expression can be rewritten using the identity for the sum of squares:
- x² + y² = (x + iy)(x – iy)
In this expression, i represents the imaginary unit, which is defined as the square root of -1. Thus, x² + y² factors into two terms: (x + iy) and (x – iy). This factorization is particularly useful when working in the complex number system.
On the other hand, if you’re dealing with sums of squares and looking for a way to handle them in real numbers, you might explore trigonometric identities or completing the square, though these approaches don’t yield a straightforward ‘factoring’ result in the traditional sense.
In summary, while x² + y² can’t be factored over the real numbers, it can be expressed in a factored form using complex numbers: (x + iy)(x – iy).