If `2x + k y + z` is a factor of `9y^(2) - z^(2) - 2xz + 6xy`, then the value of k is equal to

Options

A. `-3`
B. `-1`
C. 1
D. 3

Answer

To find the value of \( k \) such that \( 2x + ky + z \) is a factor of the expression \( 9y^2 - z^2 - 2xz + 6xy \), we can follow these steps:

### Step 1: Set up the equation
Assume that the other factor of the polynomial is of the form \( ax + by + cz \). Therefore, we can express the product of the two factors as:
\[ (2x + ky + z)(ax + by + cz) = 9y^2 - z^2 - 2xz + 6xy \]

### Step 2: Expand the left-hand side
Now, we will expand the left-hand side:
\[ (2x + ky + z)(ax + by + cz) = 2ax^2 + (2b + ak)xy + (2c + kz)xz + (kby + kz)y + (cy + bz)z \]
This gives us:
\[ = 2ax^2 + (2b + ak)xy + (2c + kz)xz + (kb)y^2 + (c + b)zy + cz^2 \]

### Step 3: Compare coefficients
Now we will compare the coefficients of the expanded expression with the original polynomial \( 9y^2 - z^2 - 2xz + 6xy \).

1. **Coefficient of \( x^2 \)**:
\[ 2a = 0 \implies a = 0 \]

2. **Coefficient of \( y^2 \)**:
\[ kb = 9 \]

3. **Coefficient of \( z^2 \)**:
\[ c = -1 \]

4. **Coefficient of \( xz \)**:
\[ 2c + kz = -2 \]

5. **Coefficient of \( xy \)**:
\[ 2b + ak = 6 \]

### Step 4: Substitute known values
From \( a = 0 \), we can simplify:
- From \( 2b + ak = 6 \):
\[ 2b = 6 \implies b = 3 \]

- From \( kb = 9 \):
\[ k \cdot 3 = 9 \implies k = 3 \]

- From \( c = -1 \):
\[ c = -1 \]

### Step 5: Conclusion
Thus, the value of \( k \) is:
\[ \boxed{3} \]