If (x-2) is a factor of the polynomial `x^3-6x^2+ax-8` then the value of a is equal to…….

Answer

To find the value of \( a \) such that \( (x-2) \) is a factor of the polynomial \( x^3 - 6x^2 + ax - 8 \), we can use the Factor Theorem. According to the Factor Theorem, if \( (x - c) \) is a factor of a polynomial \( f(x) \), then \( f(c) = 0 \).

### Step-by-Step Solution:

1. **Identify the polynomial and the factor**:
- The polynomial is \( f(x) = x^3 - 6x^2 + ax - 8 \).
- The factor given is \( (x - 2) \).

2. **Apply the Factor Theorem**:
- Since \( (x - 2) \) is a factor, we set \( x = 2 \) in the polynomial and equate it to zero:
\[ f(2) = 0 \]

3. **Substitute \( x = 2 \) into the polynomial**:
\[ f(2) = (2)^3 - 6(2)^2 + a(2) - 8 \]
- Calculate each term:
\[ f(2) = 8 - 6(4) + 2a - 8 \]
\[ f(2) = 8 - 24 + 2a - 8 \]
\[ f(2) = 2a - 24 \]

4. **Set the equation to zero**:
\[ 2a - 24 = 0 \]

5. **Solve for \( a \)**:
- Add 24 to both sides:
\[ 2a = 24 \]
- Divide by 2:
\[ a = 12 \]

### Final Answer:
The value of \( a \) is \( 12 \).

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