The repeated factor of the determinant
`|(y +z,x,y),(z +x,z,x),(x +y,y,z)|`, is

Options

A. `z -x`
B. `x -y`
C. `y -z`
D. none of these

Answer

To find the repeated factor of the determinant

\[ D = \begin{vmatrix} y + z & x & y \\ z + x & z & x \\ x + y & y & z \end{vmatrix} \]

we will perform a series of row and column operations to simplify the determinant and identify any repeated factors.

### Step 1: Add all rows to the first row
We will add the second and third rows to the first row:

\[ R_1 \rightarrow R_1 + R_2 + R_3 \]

This gives us:

\[ D = \begin{vmatrix} (y + z) + (z + x) + (x + y) & x + z + y & y + x + z \\ z + x & z & x \\ x + y & y & z \end{vmatrix} \]

Calculating the first row, we have:

\[ D = \begin{vmatrix} 2(x + y + z) & x + y + z & x + y + z \\ z + x & z & x \\ x + y & y & z \end{vmatrix} \]

### Step 2: Factor out the common term from the first row
We can factor out \(x + y + z\) from the first row:

\[ D = (x + y + z) \begin{vmatrix} 2 & 1 & 1 \\ z + x & z & x \\ x + y & y & z \end{vmatrix} \]

### Step 3: Simplify the determinant
Now we will perform column operations to simplify the determinant further:

1. \(C_1 \rightarrow C_1 - 2C_3\)
2. \(C_2 \rightarrow C_2 - C_3\)

This gives us:

\[ D = (x + y + z) \begin{vmatrix} 0 & 0 & 1 \\ z + x - 2x & z - x & x \\ x + y - 2z & y - z & z \end{vmatrix} \]

This simplifies to:

\[ D = (x + y + z) \begin{vmatrix} 0 & 0 & 1 \\ z - x & z - x & x \\ x + y - 2z & y - z & z \end{vmatrix} \]

### Step 4: Expand the determinant
Notice that the first column has two zeros, which means we can expand along the first column:

\[ D = (x + y + z) \cdot 0 = 0 \]

However, we need to check the remaining determinant:

\[ D = (x + y + z) \cdot (z - x)(y - z) \]

### Step 5: Identify repeated factors
After simplifying, we find that the determinant can be expressed as:

\[ D = (x + y + z)(z - x)(z - x) \]

Thus, the repeated factor of the determinant is:

\[ \text{Repeated Factor} = (z - x)^2 \]

### Final Answer
The repeated factor of the determinant is \(z - x\).

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