Let `f(x^(3)) = x^(4) + x^(5) + x + 1`, then the value of f(8) is

Options

A. 51
B. 52
C. 53
D. 54

Answer

To solve the problem, we need to find the value of \( f(8) \) given that \( f(x^3) = x^4 + x^5 + x + 1 \).

### Step-by-Step Solution:

1. **Understand the Function**: We are given that \( f(x^3) = x^4 + x^5 + x + 1 \). We need to find \( f(8) \).

2. **Relate \( 8 \) to \( x^3 \)**: Since we want to find \( f(8) \), we need to express \( 8 \) in the form of \( x^3 \). We can write:
\[ x^3 = 8 \]
This implies:
\[ x = 2 \quad (\text{since } 2^3 = 8) \]

3. **Substitute \( x \) into the Function**: Now that we have \( x = 2 \), we can substitute \( x \) into the function:
\[ f(8) = f(2^3) = 2^4 + 2^5 + 2 + 1 \]

4. **Calculate Each Term**:
- Calculate \( 2^4 \):
\[ 2^4 = 16 \]
- Calculate \( 2^5 \):
\[ 2^5 = 32 \]
- The other terms are \( 2 \) and \( 1 \).

5. **Add the Values Together**:
\[ f(8) = 16 + 32 + 2 + 1 \]
- First, add \( 16 + 32 \):
\[ 16 + 32 = 48 \]
- Then add \( 2 \):
\[ 48 + 2 = 50 \]
- Finally, add \( 1 \):
\[ 50 + 1 = 51 \]

6. **Final Result**: Therefore, the value of \( f(8) \) is:
\[ \boxed{51} \]