To solve the equation [tex]\((x-5)^3 + (x-8)^3 + (x-11)^3 = 3(x-5)(x-8)(x-11)\)[/tex], we use the following approach.
First, observe that for the given equation:
[tex]\[
(x-5)^3 + (x-8)^3 + (x-11)^3 = 3(x-5)(x-8)(x-11)
\][/tex]
We need to find the value of [tex]\(x\)[/tex]. Let's check if any of the potential answers [tex]\(x = 22\)[/tex], [tex]\(x = 24\)[/tex], [tex]\(x = 7\)[/tex], or [tex]\(x = 8\)[/tex] would satisfy the equation.
Checking [tex]\(x = 8\)[/tex]:
1. Substitute [tex]\(x = 8\)[/tex] into the left-hand side of the equation:
[tex]\[
(8-5)^3 + (8-8)^3 + (8-11)^3 = 3^3 + 0 + (-3)^3
\][/tex]
Simplifying each term:
[tex]\[
3^3 = 27, \quad (8-8)^3 = 0, \quad (-3)^3 = -27
\][/tex]
Thus, the left-hand side becomes:
[tex]\[
27 + 0 - 27 = 0
\][/tex]
2. Substitute [tex]\(x = 8\)[/tex] into the right-hand side of the equation:
[tex]\[
3(8-5)(8-8)(8-11) = 3 \cdot 3 \cdot 0 \cdot (-3)
\][/tex]
Simplifying:
[tex]\[
3 \times 3 \times 0 \times (-3) = 0
\][/tex]
Since both sides equal 0, [tex]\(x = 8\)[/tex] is indeed a solution.
Given [tex]\(x = 8\)[/tex] is satisfied, let's list out the other choices again:
(A) 22
(B) 24
(C) 7
(D) 8
From our calculations, the correct answer is:
[tex]\[
\boxed{8}
\][/tex]
