Certainly! Let's solve the problem of factoring the expression [tex]\(x^3 + 8a^3\)[/tex].
### Step-by-Step Solution:
1. Identify the Pattern:
The expression [tex]\(x^3 + 8a^3\)[/tex] resembles the sum of cubes, which has a standard factoring formula:
[tex]\[ A^3 + B^3 = (A + B)(A^2 - AB + B^2) \][/tex]
2. Apply the Formula:
Compare the given expression [tex]\(x^3 + 8a^3\)[/tex] with the sum of cubes formula. Here, [tex]\(A = x\)[/tex] and [tex]\(B = 2a\)[/tex].
3. Substitute into the Formula:
Using the identified values of [tex]\(A\)[/tex] and [tex]\(B\)[/tex]:
[tex]\[ (x)^3 + (2a)^3 \][/tex]
4. Factor using the Sum of Cubes Formula:
[tex]\[
\begin{align*}
A^3 + B^3 & = (A + B)(A^2 - AB + B^2) \\
x^3 + 8a^3 & = (x + 2a)((x)^2 - (x)(2a) + (2a)^2) \\
\end{align*}
\][/tex]
5. Simplify the Factors:
[tex]\[
(x + 2a)((x)^2 - (x)(2a) + (2a)^2) \\
= (x + 2a)(x^2 - 2ax + 4a^2) \\
\][/tex]
So, the factorized form of the given expression [tex]\(x^3 + 8a^3\)[/tex] is:
[tex]\[
(x + 2a)(x^2 - 2ax + 4a^2)
\][/tex]
Thus, the final factored form of the expression [tex]\(x^3 + 8a^3\)[/tex] is:
[tex]\[
(x + 2a)(x^2 - 2ax + 4a^2)
\][/tex]
