If `f(x) = |(a,-1,0),(ax,a,-1),(ax^2,ax,a)|` then f (2x) - f (x) is equal to

Options

A. ax
B. `ax(2a+3x)`
C. `ax(2+3x)`
D. none of these

Answer

To solve the problem, we need to find the expression for \( f(2x) - f(x) \) given the determinant \( f(x) = |(a, -1, 0), (ax, a, -1), (ax^2, ax, a)| \).

### Step 1: Calculate \( f(x) \)

We start with the determinant:

\[ f(x) = \begin{vmatrix} a & -1 & 0 \\ ax & a & -1 \\ ax^2 & ax & a \end{vmatrix} \]

### Step 2: Expand the determinant

Using the properties of determinants, we can expand this determinant. We can take \( a \) common from the first column:

\[ f(x) = a \begin{vmatrix} 1 & -1 & 0 \\ x & a & -1 \\ x^2 & x & a \end{vmatrix} \]

### Step 3: Expand the determinant further

Now we need to expand this determinant. We can use the first row for expansion:

\[ = a \left( 1 \cdot \begin{vmatrix} a & -1 \\ x & a \end{vmatrix} + 1 \cdot \begin{vmatrix} x & -1 \\ x^2 & a \end{vmatrix} \right) \]

Calculating the first 2x2 determinant:

\[ \begin{vmatrix} a & -1 \\ x & a \end{vmatrix} = a^2 + x \]

Calculating the second 2x2 determinant:

\[ \begin{vmatrix} x & -1 \\ x^2 & a \end{vmatrix} = ax + x^2 \]

### Step 4: Combine results

Now substituting back into our expression:

\[ f(x) = a \left( a^2 + x + ax + x^2 \right) = a(a^2 + (a + 1)x + x^2) \]

### Step 5: Calculate \( f(2x) \)

Now, we replace \( x \) with \( 2x \):

\[ f(2x) = a \left( a^2 + (a + 1)(2x) + (2x)^2 \right) = a \left( a^2 + 2(a + 1)x + 4x^2 \right) \]

### Step 6: Calculate \( f(2x) - f(x) \)

Now we find \( f(2x) - f(x) \):

\[ f(2x) - f(x) = a \left( a^2 + 2(a + 1)x + 4x^2 \right) - a \left( a^2 + (a + 1)x + x^2 \right) \]

### Step 7: Simplify the expression

Factoring out \( a \):

\[ = a \left( \left( a^2 + 2(a + 1)x + 4x^2 \right) - \left( a^2 + (a + 1)x + x^2 \right) \right) \]

This simplifies to:

\[ = a \left( (2(a + 1)x - (a + 1)x) + (4x^2 - x^2) \right) \]

\[ = a \left( (a + 1)x + 3x^2 \right) \]

### Final Result

Thus, we conclude that:

\[ f(2x) - f(x) = a \left( (a + 1)x + 3x^2 \right) \]