The value of the expression `x^(4)-17x^(3)+17x^(2)-17x+17` at x=16 is :

Options

A. 0
B. 1
C. 33
D. 15

Answer

To find the value of the expression \( x^4 - 17x^3 + 17x^2 - 17x + 17 \) at \( x = 16 \), we will substitute \( x \) with \( 16 \) and simplify the expression step by step.

### Step-by-Step Solution:

1. **Substitute \( x = 16 \) into the expression:**
\[ 16^4 - 17 \cdot 16^3 + 17 \cdot 16^2 - 17 \cdot 16 + 17 \]

2. **Calculate \( 16^4 \):**
\[ 16^4 = 65536 \]

3. **Calculate \( 17 \cdot 16^3 \):**
\[ 16^3 = 4096 \quad \text{so} \quad 17 \cdot 16^3 = 17 \cdot 4096 = 69632 \]

4. **Calculate \( 17 \cdot 16^2 \):**
\[ 16^2 = 256 \quad \text{so} \quad 17 \cdot 16^2 = 17 \cdot 256 = 4352 \]

5. **Calculate \( 17 \cdot 16 \):**
\[ 17 \cdot 16 = 272 \]

6. **Now substitute these values back into the expression:**
\[ 65536 - 69632 + 4352 - 272 + 17 \]

7. **Perform the calculations step by step:**
- First, calculate \( 65536 - 69632 \):
\[ 65536 - 69632 = -4096 \]
- Next, add \( 4352 \):
\[ -4096 + 4352 = 256 \]
- Then, subtract \( 272 \):
\[ 256 - 272 = -16 \]
- Finally, add \( 17 \):
\[ -16 + 17 = 1 \]

8. **Final Result:**
\[ \text{The value of the expression at } x = 16 \text{ is } 1. \]

### Conclusion:
The value of the expression \( x^4 - 17x^3 + 17x^2 - 17x + 17 \) at \( x = 16 \) is \( \boxed{1} \).