Question

Which expression(s) have a greatest common factor (GCF) of 3xy2 with 42xy4?
Select all that apply.
A. 6x2y2
B. 15xy2z
C. 27x3y2
D. 9xy3

Answer 1

To determine which expressions have a greatest common factor (GCF) of \(3xy^2\) with \(42xy^4\), we can follow these steps:

1. **Factor 42xy^4:**
We start by factoring \(42xy^4\):
\[
42 = 2 \times 3 \times 7
\]
Therefore:
\[
42xy^4 = 2 \times 3 \times 7 \times x \times y^4
\]

2. **Identify the GCF**:
The GCF should be \(3xy^2\). This means the numerical coefficient must have a GCF of 3, and the variables involved must be \(x\) with an exponent of at least 1 and \(y\) with an exponent of at least 2.

3. **Analyze each expression**:

A. **\(6x^2y^2\)**:
- Coefficient: \(6\), factors are \(2 \times 3\) → GCF with \(42\) is \(3\).
- Variables: \(x^2\) (is ≥ 1) and \(y^2\) (is ≥ 2).
- GCF matches \(3xy^2\).

B. **\(15xy^2z\)**:
- Coefficient: \(15\), factors are \(3 \times 5\) → GCF with \(42\) is \(3\).
- Variables: \(x\) (is ≥ 1) and \(y^2\) (is ≥ 2).
- GCF matches \(3xy^2\).

C. **\(27x^3y^2\)**:
- Coefficient: \(27\), factors are \(3^3\) → GCF with \(42\) is \(3\).
- Variables: \(x^3\) (is ≥ 1) and \(y^2\) (is ≥ 2).
- GCF matches \(3xy^2\).

D. **\(9xy^3\)**:
- Coefficient: \(9\), factors are \(3^2\) → GCF with \(42\) is \(3\).
- Variables: \(x\) (is ≥ 1) and \(y^3\) (is ≥ 2).
- GCF matches \(3xy^2\).

4. **Conclusion**:
All four expressions share a GCF of \(3xy^2\) with \(42xy^4\).

Thus, the correct choices are:
- A. \(6x^2y^2\)
- B. \(15xy^2z\)
- C. \(27x^3y^2\)
- D. \(9xy^3\)