We have to factorize x^6 - 64

x^6 - 64

=> (x^3)^2 - 8^2

=> (x^3 - 8)(x^3 + 8)

=> (x^3 - 2^3)(x^3 + 8)

=> (x - 2)(x^2 + 2x + 4)(x^3 + 8)

**The factors of x^6 - 64 = (x - 2)(x^2 + 2x + 4)(x^3 + 8)**

The expression x^6 – 64 has to be factorized.

Use the formula a^2 - b^2 = (a - b)(a + b)

x^6 = x^(3*2) = (x^3)^2

x^6 - 64

= (x^3)^2 - 8^2

= (x^3 - 8)(x^3 + 8)

Now use the formula a^3 - b^3 = (a-b)(a^2 + ab + b^2).

(x^3 - 8)(x^3 + 8)

= (x^3 - 2^3)(x^3 + 8)

= (x - 2)(x^2 + 2x + 4)(x^3 + 8)

The factored form of x^6 – 64 is (x - 2)(x^2 + 2x + 4)(x^3 + 8)

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)