Which one of the following factors does the expansions of the determinant `((x,y,3),(x^(2),5y^(2),9),(x^(3),10y^(3),27))` contain ?

Options

A. x - 3
B. x - y
C. y - 3
D. x - 3y

Answer

To solve the problem of finding which factor the expansion of the determinant

\[ \begin{vmatrix} x & y & 3 \\ x^2 & 5y^2 & 9 \\ x^3 & 10y^3 & 27 \end{vmatrix} \]

contains, we will follow these steps:

### Step 1: Write the Determinant
We start by writing the determinant in a clear format:

\[ D = \begin{vmatrix} x & y & 3 \\ x^2 & 5y^2 & 9 \\ x^3 & 10y^3 & 27 \end{vmatrix} \]

### Step 2: Apply Column Operations
Next, we will perform column operations to simplify the determinant. We can perform the operation \( C_3 \leftarrow C_3 - C_1 \):

\[ D = \begin{vmatrix} x & y & 3 - x \\ x^2 & 5y^2 & 9 - x^2 \\ x^3 & 10y^3 & 27 - x^3 \end{vmatrix} \]

### Step 3: Simplify the Third Column
Now, we simplify the third column:

- The first row becomes \( 3 - x \).
- The second row becomes \( 9 - x^2 = (3 - x)(3 + x) \).
- The third row becomes \( 27 - x^3 = (3 - x)(9 + 3x + x^2) \).

Thus, we can rewrite the determinant as:

\[ D = \begin{vmatrix} x & y & 3 - x \\ x^2 & 5y^2 & (3 - x)(3 + x) \\ x^3 & 10y^3 & (3 - x)(9 + 3x + x^2) \end{vmatrix} \]

### Step 4: Factor Out \( (3 - x) \)
Since \( (3 - x) \) is common in the third column, we can factor it out:

\[ D = (3 - x) \begin{vmatrix} x & y & 1 \\ x^2 & 5y^2 & 3 + x \\ x^3 & 10y^3 & 9 + 3x + x^2 \end{vmatrix} \]

### Step 5: Identify the Factor
From the above steps, we see that the factor \( (3 - x) \) is present in the expansion of the determinant.

Thus, the factor contained in the expansion of the determinant is:

\[ \boxed{(x - 3)} \]