If `f(x)=x^3=x^2+a x+b` is divisible by `x^2-x` , then find the value of `f(2)dot`

Explanation

To solve the problem, we need to find the value of \( f(2) \) given that the polynomial \( f(x) = x^3 - x^2 + ax + b \) is divisible by \( x^2 - x \).

### Step 1: Understand the Divisibility Condition
Since \( f(x) \) is divisible by \( x^2 - x \), it means that both \( x = 0 \) and \( x = 1 \) are roots of \( f(x) \).

### Step 2: Apply the Root Condition at \( x = 0 \)
Substituting \( x = 0 \) into \( f(x) \):
\[ f(0) = 0^3 - 0^2 + a(0) + b = b \]
Since \( f(0) = 0 \), we have:
\[ b = 0 \]

### Step 3: Apply the Root Condition at \( x = 1 \)
Now substituting \( x = 1 \) into \( f(x) \):
\[ f(1) = 1^3 - 1^2 + a(1) + b = 1 - 1 + a + b = a + b \]
Since \( f(1) = 0 \) and we already found that \( b = 0 \), we have:
\[ a + 0 = 0 \implies a = 0 \]

### Step 4: Substitute Values of \( a \) and \( b \) into \( f(x) \)
Now substituting \( a \) and \( b \) back into the function:
\[ f(x) = x^3 - x^2 + 0 \cdot x + 0 = x^3 - x^2 \]

### Step 5: Calculate \( f(2) \)
Now we need to find \( f(2) \):
\[ f(2) = 2^3 - 2^2 = 8 - 4 = 4 \]

### Final Answer
Thus, the value of \( f(2) \) is:
\[ \boxed{4} \]