To find the completely factored form of [tex]\(x^3 - 64x\)[/tex], we follow these steps:
1. Identify Common Factor: Notice that [tex]\(x\)[/tex] is a common factor in each term of the expression [tex]\(x^3 - 64x\)[/tex]. We can factor out [tex]\(x\)[/tex] from the whole expression:
[tex]\[
x^3 - 64x = x(x^2 - 64)
\][/tex]
2. Factor the Quadratic Expression: Next, we need to factor [tex]\(x^2 - 64\)[/tex]. Recognize that [tex]\(x^2 - 64\)[/tex] is a difference of squares. Recall that the difference of squares can be factored using the formula:
[tex]\[
a^2 - b^2 = (a - b)(a + b)
\][/tex]
Here, we have [tex]\(a = x\)[/tex] and [tex]\(b = 8\)[/tex], so:
[tex]\[
x^2 - 64 = x^2 - 8^2 = (x - 8)(x + 8)
\][/tex]
3. Combine the Factors: Substitute back the factors of [tex]\(x^2 - 64\)[/tex] into the expression we factored out previously:
[tex]\[
x(x^2 - 64) = x(x - 8)(x + 8)
\][/tex]
Therefore, the completely factored form of [tex]\(x^3 - 64x\)[/tex] is:
[tex]\[
x(x - 8)(x + 8)
\][/tex]
Among the given options, the correct answer is:
[tex]\[
x(x-8)(x+8)
\][/tex]
