To rewrite the expression [tex]\(x^3 - 64\)[/tex] using the difference of cubes, we need to recognize that 64 can be expressed as a power of 4, specifically [tex]\(4^3\)[/tex]. The difference of cubes formula is:
[tex]\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \][/tex]
In the given expression [tex]\(x^3 - 64\)[/tex], we can set [tex]\(a = x\)[/tex] and [tex]\(b = 4\)[/tex] since [tex]\(64 = 4^3\)[/tex].
Substituting [tex]\(a\)[/tex] and [tex]\(b\)[/tex] into the formula, we get:
[tex]\[ x^3 - 64 = x^3 - 4^3 \][/tex]
Now, applying the difference of cubes formula:
[tex]\[ x^3 - 4^3 = (x - 4)(x^2 + 4x + 16) \][/tex]
We need to compare this with the given options to identify the correct factorization:
A. [tex]\((x+4)(x^2 - 4x + 16)\)[/tex]
B. [tex]\((x-4)(x^2 + 4x + 16)\)[/tex]
C. [tex]\((x-4)(x^2 + 16x + 4)\)[/tex]
D. [tex]\((x+4)(x^2 - 4x - 16)\)[/tex]
Looking at the options, the correct factorization matches option B:
[tex]\[ (x-4)(x^2 + 4x + 16) \][/tex]
Therefore, the correct answer is:
B. [tex]\((x-4)(x^2 + 4x + 16)\)[/tex]
