Factorize x^6 – 64.

We could write 64 as a power of
2:

64 = 2^6

Now, we can apply the
formula:

x^n - a^n = (x-a)(x^(n-1) + x^(n-2)*a + .... +
a^(n-1))

We'll put n = 6 and a =
2:

x^6 - 2^6 = (x-2)(x^5 + 2x^4 + 4x^3 + 8x^2 + 16x +
32)

We also could
write:

(x^2)^3 - (2^2)^3

a^3 - b^3 =
(a-b)(a^2 + ab + b^2)

We'll put a = x^2 and b =
2^2

(x^2)^3 - (2^2)^3 = (x^2 - 4)(x^4 + 4x^2 +
16)

But x^2 - 4 is a difference of
squares:

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

(x^2)^3 -
(2^2)^3 = (x-2)(x+2)(x^4 + 4x^2 + 16)

x^6 - 2^6 =
(x-2)(x+2)(x^4 + 4x^2 + 16)