To factor the expression [tex]\(x^3 + 8\)[/tex], we can recognize it as a sum of cubes. The general formula for factoring a sum of cubes is:
[tex]\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \][/tex]
In our case, we have [tex]\(x^3 + 8\)[/tex]. Notice that [tex]\(8\)[/tex] can be written as [tex]\(2^3\)[/tex]. So, we identify [tex]\(a\)[/tex] as [tex]\(x\)[/tex] and [tex]\(b\)[/tex] as [tex]\(2\)[/tex]. Applying the sum of cubes formula, we substitute [tex]\(a = x\)[/tex] and [tex]\(b = 2\)[/tex]:
[tex]\[
x^3 + 2^3 = (x + 2)(x^2 - 2x + 4)
\][/tex]
Let's break it down step-by-step:
1. Identify the cubes:
[tex]\[
a = x \quad \text{and} \quad b = 2
\][/tex]
2. Substitute into the sum of cubes formula:
[tex]\[
(a + b)(a^2 - ab + b^2) = (x + 2)((x)^2 - (x)(2) + (2)^2)
\][/tex]
3. Simplify the expression inside the parentheses:
[tex]\[
(x + 2)(x^2 - 2x + 4)
\][/tex]
Therefore, the factored form of the expression [tex]\(x^3 + 8\)[/tex] is:
[tex]\[
(x + 2)(x^2 - 2x + 4)
\][/tex]
