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