If x-2 is a factor of `x^(2)-7x+2a`, find the value of a.

Answer

To find the value of \( a \) such that \( x - 2 \) is a factor of the polynomial \( x^2 - 7x + 2a \), we can follow these steps:

### Step 1: Set up the polynomial
We have the polynomial:
\[ P(x) = x^2 - 7x + 2a \]

### Step 2: Use the factor theorem
According to the factor theorem, if \( x - 2 \) is a factor of \( P(x) \), then \( P(2) = 0 \).

### Step 3: Substitute \( x = 2 \) into the polynomial
Now, we substitute \( x = 2 \) into the polynomial:
\[ P(2) = 2^2 - 7(2) + 2a \]

### Step 4: Simplify the expression
Calculating \( P(2) \):
\[ P(2) = 4 - 14 + 2a \]
\[ P(2) = -10 + 2a \]

### Step 5: Set the polynomial equal to zero
Since \( P(2) = 0 \):
\[ -10 + 2a = 0 \]

### Step 6: Solve for \( a \)
Now, we solve for \( a \):
\[ 2a = 10 \]
\[ a = \frac{10}{2} = 5 \]

### Conclusion
Thus, the value of \( a \) is:
\[ \boxed{5} \]

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