Factor x^2+y^2

Problem

x2+y2

Solution

1. Identify the expression as a sum of squares, which does not factor over the set of real numbers.

2. Apply the complex number identity a2+b2=(a+b*i)*(a−b*i) to factor the expression over the complex field.

3. Substitute x for a and y for b into the identity.

4. Verify the result by expanding (x+i*y)*(x−i*y)=x2−i*x*y+i*x*y−i2*y2=x2−(−1)*y2=x2+y2

Final Answer

x2+y2=(x+i*y)*(x−i*y)

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