If `f(x)=|a-1 0a x a-1a x^2a x a|` , using properties of determinants, find the value of `f(2x)-f(x)` .

Explanation

To solve the problem, we need to evaluate the expression \( f(2x) - f(x) \) where \( f(x) \) is defined as the determinant:

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

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

We will first compute the determinant \( f(x) \).

Using the properties of determinants, we can perform row operations. We will transform \( R_3 \) to \( R_2 \) by replacing \( R_3 \) with \( R_3 - R_1 \):

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

Now we can expand the determinant along the first row:

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

Calculating the first determinant:

\[ \begin{vmatrix} a-1 & ax \\ ax^2 + 1 & a + 1 \end{vmatrix} = (a-1)(a+1) - ax(ax^2 + 1) = a^2 - 1 - a^2x^3 - ax \]

Calculating the second determinant:

\[ \begin{vmatrix} ax & ax \\ 0 & a + 1 \end{vmatrix} = ax(a + 1) - ax(0) = ax(a + 1) \]

Putting it all together, we have:

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

### Step 2: Simplify \( f(x) \)

Now we can simplify \( f(x) \):

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

### Step 3: Calculate \( f(2x) \)

Now we calculate \( f(2x) \):

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

Following similar steps as above, we can compute \( f(2x) \):

\[ = a(a^2 - 1 - a^2(2x)^3 - a(2x)) + a(2x)(a + 1) \]
\[ = a(a^2 - 1 - 8a^2x^3 - 2ax) + 2ax(a + 1) \]

### Step 4: Calculate \( f(2x) - f(x) \)

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

\[ f(2x) - f(x) = \left[ a(a^2 - 1 - 8a^2x^3 - 2ax) + 2ax(a + 1) \right] - \left[ a(a^2 - 1 - a^2x^3 - ax) \right] \]

Simplifying this expression will yield the final result.

### Final Result

After simplifying, we find:

\[ f(2x) - f(x) = a(2a + 3x) \]