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Joshua asks, "I heard people say that a sum of two squares is (x + y + sqr(2xy))(x + y - sqr(2xy)) But I also heard that a sum of two squares is (x+iy)(x-iy) Are both of these correct?? And are there other ways to factorize the sum of two perfect squares?"
Hi Joshua, the process of factorization is the process of breaking down an expression into two or more expressions which, when multiplied together, give the original expression. So is it possible to factor something in more than one way? It surely is. Consider the following expression: 12. I know, it's a boring expression; it doesn't even have any variables. But it's still an expression, and it can be factored in many ways: 1 x 12; 2 x 6; 3 x 4; 2 x 2 x 3; -2 x -6 ... it can even be factored as 0.5 x 24. Usually when we're talking about factoring numbers, we think about integers, but our definition doesn't require that. So yes, there may be multiple ways to factor an algebraic expression. Some of them will be prettier than others, and sometimes the factorizations we find may not be at all useful, but that doesn't change the fact that they exist. Typically, when we talk about factorizations, we're looking for polynomial factors with real coefficients, and neither of the factorizations you gave fit that description, but they're still factorizations if they multply to the given expression.
So with that as background, let's take a look at your two expressions and see if they really are factorizations of x2 + y2. The best way to do that is to multiply the factors together and see what happens.
(x + y + √(2xy))(x + y - √(2xy))
x((x + y - √(2xy)) + y((x + y - √(2xy)) + √(2xy)((x + y - √(2xy))
(x2 + xy - x√(2xy)) + (yx + y2 - y√(2xy)) + (x√(2xy) + y√(2xy) - 2xy)
x2 + y2
So, yes, that is a valid factorization of x2 + y2. Let's try the other.
(x + iy)(x - iy)
x( x - iy) + iy(x - iy)
x2 - ixy +iyx - (-y2)
x2 + y2
Is there another factorization? Can we find it? Sure we can! Here are a couple possibilities:
x2 + y2 = x2(1 + (y/x)2)
x2 + y2 = (√(x2 + y2 + 4) + 2)((√(x2 + y2 + 4) - 2)
Again, these are not likely to be useful factorizations, but they still fit the definition of a factorization. In closing, I should point out that if, as I mentioned earlier, we're looking for polynomial factors with real coefficients, then x2 + y2 is irreducible, which is a fancy way of saying that it can be written only one way:
x2 + y2 = 1(x2 + y2).