Let's analyze the given expression, [tex]\(3x^3 + 24\)[/tex], step by step.
1. Factor out the Greatest Common Factor (GCF):
First, we need to identify the GCF of the terms in the expression [tex]\(3x^3 + 24\)[/tex]. The GCF of [tex]\(3x^3\)[/tex] and [tex]\(24\)[/tex] is [tex]\(3\)[/tex].
So, we factor out [tex]\(3\)[/tex] from the expression:
[tex]\[
3x^3 + 24 = 3(x^3 + 8)
\][/tex]
2. Analyze the factored expression:
Now, we consider the expression inside the parentheses: [tex]\(x^3 + 8\)[/tex].
Notice that [tex]\(8\)[/tex] can be written as [tex]\(2^3\)[/tex]. Hence, we can rewrite the expression as:
[tex]\[
x^3 + 8 = x^3 + 2^3
\][/tex]
3. Identify if it is a sum of cubes:
The expression [tex]\(x^3 + 2^3\)[/tex] is indeed a sum of cubes. The sum of cubes can be factored using the formula:
[tex]\[
a^3 + b^3 = (a+b)(a^2 - ab + b^2)
\][/tex]
Here, [tex]\(a = x\)[/tex] and [tex]\(b = 2\)[/tex].
4. Apply the sum of cubes formula:
Using the sum of cubes formula, we get:
[tex]\[
x^3 + 2^3 = (x + 2)(x^2 - 2x + 4)
\][/tex]
5. Write the fully factored expression:
Now, substituting back into our factored-out GCF, we have:
[tex]\[
3(x^3 + 8) = 3(x + 2)(x^2 - 2x + 4)
\][/tex]
So, the expression [tex]\(3x^3 + 24\)[/tex] can indeed be factored as a sum of cubes, and the fully factored form is:
[tex]\[
3(x + 2)(x^2 - 2x + 4)
\][/tex]
