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The Sum of Two Squares

Do you think x^4+4 is factorable?

In general, the sum of two squares is not factorable, such as x^2+1. However, there are special situations in which the sum of two squares is factorable, like the one in our title.

You don't believe it? Let's try it.

 

x^4+4

The trick here is to add an extra term to this polynomial, with a purpose of completing the square.

Usually, when we complete the square, we will add a constant term to a quadratic expression, and subtract that same term at the end of the expression. But this time, we are adding the "middle" term. (Of course, we will need to subtract it at the end so that we don't change our original expression.)

x^4 + 4x^2 +4 - 4x^2

Here we can group the first three terms of the expression:

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

Write the grouped terms as a perfect square:

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

Now, do you recognize this is the difference of squares?

Difference of Squares
 

a^2-b^2=(a+b)(a-b)

To use the difference of two squares formula, we can write (x^2+2)^2-4x^2 as:

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

Using the difference of two squares formula, we get:

(x^2+2)^2-(2x)^2 ~doubleright~ (x^2+2+2x)(x^2+2-2x)
 

Write each factor of the product in descending order:

~right~(x^2+2x+2)(x^2-2x+2)