Simplifying Quadratic equestions

[dreamz008]

Why does f^8-g^8= (f^4+g^4)(f^2+g^2)(f+g)(f-g)?

Can you please explain this step by step so that I can identify where I am making an error?

[JonathanSchneider]

So that I don't need to keep using carrots, I am going to henceforth refer to f^8 simply as f8 and g^8 as g8, etc.

f8 - g8 is the difference between two perfect squares. (they are the squares of f4 and g4.) thus, the expression can be rewritten this way:

(f4 + g4)(f4 - g4)

Now, f4 - g4 is ALSO the difference between two perfect squares, namely f2 and g2. Thus, the latter part of the above expression can also be rewritten. Our next line:

(f4 + g4)(f2 + g2)(f2 - g2)

Finally, f2 - g2 is the again the difference between two perfect squares. Thus, we have:

(f4 + g4)(f2 + g2)(f + g)(f - g)