If a,b,c are real `x^(3)-3b^(2)x+2c^(3)` is divisible by x-a and x-b, then

Options

A. `a=-b =-c`
B. `a=2b=2c`
C. `a=b=c" or "a=-2b=-2c`
D. none of these

Answer

To solve the problem, we need to determine the conditions under which the polynomial \( P(x) = x^3 - 3b^2x + 2c^3 \) is divisible by both \( x - a \) and \( x - b \). This means that both \( a \) and \( b \) are roots of the polynomial.

### Step 1: Set up the equations for the roots
Since \( P(a) = 0 \) and \( P(b) = 0 \), we can write the following equations:

1. \( P(a) = a^3 - 3b^2a + 2c^3 = 0 \) (Equation 1)
2. \( P(b) = b^3 - 3b^2b + 2c^3 = 0 \) (Equation 2)

### Step 2: Simplify Equation 2
From Equation 2, we simplify:

\[ b^3 - 3b^3 + 2c^3 = 0 \]
\[ -2b^3 + 2c^3 = 0 \]
\[ 2c^3 = 2b^3 \]
\[ c^3 = b^3 \]

Taking the cube root of both sides, we find:

\[ c = b \quad \text{(since } a, b, c \text{ are real)} \]

### Step 3: Substitute \( c = b \) into Equation 1
Now substitute \( c = b \) into Equation 1:

\[ a^3 - 3b^2a + 2b^3 = 0 \]

### Step 4: Rearranging Equation 1
Rearranging gives us:

\[ a^3 - 3b^2a + 2b^3 = 0 \]

### Step 5: Check for specific values of \( a \)
We can check if specific values of \( a \) satisfy this equation.

1. **Check \( a = b \)**:
\[ b^3 - 3b^2b + 2b^3 = 0 \]
\[ b^3 - 3b^3 + 2b^3 = 0 \]
\[ 0 = 0 \quad \text{(True)} \]

2. **Check \( a = -b \)**:
\[ (-b)^3 - 3b^2(-b) + 2b^3 = 0 \]
\[ -b^3 + 3b^3 + 2b^3 = 0 \]
\[ 4b^3 = 0 \quad \text{(Only true if } b = 0) \]

3. **Check \( a = 2b \)**:
\[ (2b)^3 - 3b^2(2b) + 2b^3 = 0 \]
\[ 8b^3 - 6b^3 + 2b^3 = 0 \]
\[ 4b^3 = 0 \quad \text{(Only true if } b = 0) \]

### Conclusion
The values of \( a \) that satisfy the condition are \( a = b \) and potentially \( a = -b \) or \( a = 2b \) under specific conditions.