When `2x^(3) + 5x^(2)- 2x + 8` is divided by `(x-a)` the remainder is `2a^(3) + 5a^(2)`. Find the value of a.

Answer

To solve the problem, we need to find the value of \( a \) such that when the polynomial \( 2x^3 + 5x^2 - 2x + 8 \) is divided by \( (x - a) \), the remainder is \( 2a^3 + 5a^2 \).

### Step 1: Understand the Remainder Theorem
According to the Remainder Theorem, when a polynomial \( P(x) \) is divided by \( (x - a) \), the remainder is \( P(a) \).

### Step 2: Define the Polynomial
Let \( P(x) = 2x^3 + 5x^2 - 2x + 8 \).

### Step 3: Calculate \( P(a) \)
Now, we need to calculate \( P(a) \):
\[ P(a) = 2a^3 + 5a^2 - 2a + 8 \]

### Step 4: Set Up the Equation
According to the problem, the remainder when dividing by \( (x - a) \) is given as \( 2a^3 + 5a^2 \). Therefore, we can set up the equation:
\[ P(a) = 2a^3 + 5a^2 \]
This gives us:
\[ 2a^3 + 5a^2 - 2a + 8 = 2a^3 + 5a^2 \]

### Step 5: Simplify the Equation
Now, we can simplify the equation by subtracting \( 2a^3 + 5a^2 \) from both sides:
\[ 2a^3 + 5a^2 - 2a + 8 - (2a^3 + 5a^2) = 0 \]
This simplifies to:
\[ -2a + 8 = 0 \]

### Step 6: Solve for \( a \)
Now, we can solve for \( a \):
\[ -2a + 8 = 0 \implies -2a = -8 \implies a = 4 \]

### Conclusion
Thus, the value of \( a \) is \( 4 \).

---