Find the value of a, if x -2 is a factor of `2x^5- 6x^4 -2ax^3+ 6ax^2+ 4ax + 8.`

Answer

To find the value of \( a \) such that \( x - 2 \) is a factor of the polynomial \( 2x^5 - 6x^4 - 2ax^3 + 6ax^2 + 4ax + 8 \), we can use the Remainder Theorem. According to the theorem, if \( x - c \) is a factor of a polynomial \( P(x) \), then \( P(c) = 0 \).

### Step-by-Step Solution:

1. **Identify the polynomial**:
\[ P(x) = 2x^5 - 6x^4 - 2ax^3 + 6ax^2 + 4ax + 8 \]

2. **Set the value of \( x \)**:
Since \( x - 2 \) is a factor, we will substitute \( x = 2 \) into the polynomial.

3. **Substitute \( x = 2 \) into \( P(x) \)**:
\[ P(2) = 2(2^5) - 6(2^4) - 2a(2^3) + 6a(2^2) + 4a(2) + 8 \]

4. **Calculate powers of 2**:
\[ 2^5 = 32, \quad 2^4 = 16, \quad 2^3 = 8, \quad 2^2 = 4 \]

5. **Substitute these values into the polynomial**:
\[ P(2) = 2(32) - 6(16) - 2a(8) + 6a(4) + 4a(2) + 8 \]

6. **Simplify the expression**:
\[ P(2) = 64 - 96 - 16a + 24a + 8 \]
\[ P(2) = 64 + 8 - 96 + 8a \]
\[ P(2) = -24 + 8a \]

7. **Set \( P(2) = 0 \)**:
\[ -24 + 8a = 0 \]

8. **Solve for \( a \)**:
\[ 8a = 24 \]
\[ a = \frac{24}{8} = 3 \]

### Final Answer:
The value of \( a \) is \( 3 \).