So for example if we have this expression(Fig. A):
I can conclude in blink of an eye that it can be factorized into (Fig. B), but, how do I proove that?
Simple! just multiply the parenthesis and it should give you (Fig C.), then the two "xy" cancell each other and we have just with what we started off(Fig. D) We just turned two factors into one!
Remember that Factorizing's purpose is to turn an x factor expression into the lowest, one if possible, quantity of factors.
So in easy words you just have to take the square root of each of the terms and add them in two final parenthesis in this form: (Square root of term1 + Square root of term2)(Square root of term1 - Square root of term2). Too easy, in fact.
Special Cases:Sometimes we can get that the expression, instead of being squared, it is powered to 4, which, still makes it possble to do, but how(Fig. A)? Simple, remember that we have to get the square roots of each term, so first, the square root of x^4 is x^2, then y^4 is y^2, so we form the parenthesis and, we are done(Fig B.)! Not quite so! I just found another expression that can still be factorized even more, did you notice it? it is the (x^2-y^2), which is a difference of squares, factorize as usual and it would give you (Fig. C). The same process is for powers which are powers of 2, for example (4, 8, 16, 32, etc..)
