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Believe It or not, sometimes the Sum of Two Squares Can Be Factored

The sum of two squares is mostly regarded as prime. Nevertheless, there are special cases where it does factor. To be eligible, one of the terms in the binomial must be an even power greater than 2 (For instance, x^4, x^6, and x^48).

The simplest sum of two squares that can be factored is x^4 + 4. It is the sum of two squares because it can be expressed as (x^2)^2 + 2^2.

To factor it, we must first tweak it a little bit, as to turn part of the expression into a perfect square trinomial.

Recall that (A + B)^2 = A^2 + B^2 + 2AB, where A and B are real numbers.

Now let's take the two bases of the sum we're trying to factor and equal them to A and B. Thus we have A = x^2 and B = 2.

Notice that we still don't have a 2AB term, or in our case 2*(x^2)*2 = 4x^2

The trick is to add that term to our expression and then subtract it at the same time, leaving us with x^4 + 4 + 4x^2 - 4x^2.

It's now clear that the first three terms make up a perfect square trinomial, which factors as (x^2 + 2)^2. So after we factor it, we get (x^2 + 2)^2 - 4x^2.

And what is special about this last result? Well, it is nothing more than THE DIFFERENCE OF TWO SQUARES, which factors out as:

(x^2 + 2x + 2)(x^2 - 2x +2).

Therefore x^4 + 4 = (x^2 + 2x + 2)(x^2 - 2x +2)

If you don't believe me, distribute it.

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