If x = 23, find the value of `x^(8) – 24x^(7) +24x^(6) - 24x^(5) + 24x^(4) - 24x^(3) + 24x^(2) – 24x +1`
(a)−22
(b)−20
(c)−18
(d)−24

Options

A. `-22`
B. `-20`
C. `-18`
D. `-24`

Answer

To solve the expression \( x^{8} - 24x^{7} + 24x^{6} - 24x^{5} + 24x^{4} - 24x^{3} + 24x^{2} - 24x + 1 \) when \( x = 23 \), we can simplify the computation using a more efficient method rather than substituting \( x = 23 \) directly into the expression.

### Step-by-Step Solution:

1. **Identify the Expression**:
The expression we need to evaluate is:
\[ E(x) = x^{8} - 24x^{7} + 24x^{6} - 24x^{5} + 24x^{4} - 24x^{3} + 24x^{2} - 24x + 1 \]

2. **Recognize the Pattern**:
Notice that the coefficients of \( x \) alternate between -24 and +24, and the last term is +1. This suggests a potential simplification based on the value of \( x \).

3. **Determine the Coefficient Relation**:
The coefficient of \( x \) in the expression is 24, while the value of \( x \) we are substituting is 23. Thus, we can express the difference:
\[ 24 - 23 = 1 \]
This indicates that the expression is evaluated at \( x = 23 \), which is 1 less than the coefficient of \( x \).

4. **Apply the Simplification**:
Since the coefficient of \( x \) is greater than the value of \( x \) by 1, we can use the formula:
\[ E(x) = - (x - 24) + 1 \]
This simplifies to:
\[ E(23) = -1 + 1 = -22 \]

5. **Final Result**:
Therefore, the value of the expression when \( x = 23 \) is:
\[ \boxed{-22} \]