Write the greatest common factor in each of the term.
`3x ^(2) y, 18 x y ^(), - 6 xy `

Options

A. `3x^2y`
B. `-3xy`
C. `3xy`
D. `6xy`

Answer

To find the greatest common factor (GCF) of the terms \(3x^2y\), \(18xy\), and \(-6xy\), we will follow these steps:

### Step 1: Factor each term
1. **First term**: \(3x^2y\)
- This can be factored as \(3 \cdot x \cdot x \cdot y\) or \(3 \cdot x^2 \cdot y\).

2. **Second term**: \(18xy\)
- This can be factored as \(18 = 3 \cdot 6 = 3 \cdot 2 \cdot 3\), so we can write it as \(3 \cdot 6 \cdot x \cdot y\) or \(3 \cdot 2 \cdot 3 \cdot x \cdot y = 3 \cdot 3 \cdot 2 \cdot x \cdot y\).

3. **Third term**: \(-6xy\)
- This can be factored as \(-1 \cdot 6xy = -1 \cdot (3 \cdot 2) \cdot x \cdot y = -1 \cdot 3 \cdot 2 \cdot x \cdot y\).

### Step 2: Identify the common factors
Now we will identify the common factors from the factored forms:
- From \(3x^2y\): \(3\), \(x\), \(y\)
- From \(18xy\): \(3\), \(2\), \(x\), \(y\)
- From \(-6xy\): \(-1\), \(3\), \(2\), \(x\), \(y\)

### Step 3: Determine the GCF
The common factors in all three terms are:
- The number \(3\) is common in all three terms.
- The variable \(x\) is common in the first and second terms, but not in the third term.
- The variable \(y\) is common in all three terms.

Thus, the greatest common factor (GCF) is:
\[ GCF = 3 \cdot y = 3y \]

### Step 4: Write the GCF
The greatest common factor of the terms \(3x^2y\), \(18xy\), and \(-6xy\) is:
\[ \text{GCF} = 3y \]