Factor out the CGF

Question

3, x, squared, y, plus, 15, x, y, minus, 21, x, y, squared3x
2
y+15xy−21xy
2

Answer
Attempt 1 out of 2
Multiple Choice Answers
3, x, left parenthesis, x, y, plus, 5, y, minus, 7, y, right parenthesis3x(xy+5y−7y)
3, x, y, left parenthesis, x, plus, 5, minus, 7, y, right parenthesis3xy(x+5−7y)
3, x, left parenthesis, x, y, plus, 5, y, minus, 7, y, squared, right parenthesis3x(xy+5y−7y
2
)
3, x, y, left parenthesis, 1, plus, 5, minus, 7, y, right parenthesis3xy(1+5−7y)

Answer 1

To factor the expression \(3x^2y + 15xy - 21xy^2\), we first look for the greatest common factor (GCF) of the coefficients and variables in the terms.

1. Identify the GCF:
   • The coefficients are \(3\), \(15\), and \(-21\). The GCF of these numbers is \(3\).
   • The variable part consists of \(x^2y\), \(xy\), and \(-21xy^2\). The GCF for the variable part is \(xy\), as all terms have at least one \(x\) and one \(y\).

Thus, the overall GCF is \(3xy\).

2. Factor out the GCF: Now we divide each term in the expression by the GCF \(3xy\): \[ \frac{3x^2y}{3xy} + \frac{15xy}{3xy} - \frac{21xy^2}{3xy} = x + 5 - 7y \]

Therefore, we can rewrite the expression as: \[ 3xy(x + 5 - 7y) \]

3. Conclusion: The factored form of the expression \(3x^2y + 15xy - 21xy^2\) is: \[ 3xy(x + 5 - 7y) \]

Among the provided answer choices, this corresponds to:

• \(3xy(x + 5 - 7y)\) or it's equivalent.

However, this doesn’t seem to appear directly in the answer choices given. Let’s double-check the corresponding choice:

• The closest choice would be \(3x\left(xy + 5y - 7y^2\right)\) which results by grouping differently and factoring the coefficients accordingly, taking care of terms.

Thus, the correct interpretation of the multiple-choice may point to: \[ 3x\left(xy + 5y - 7y^2\right) \]

This matches up with looking for \(3xy\)’s common factors.

Final answer: \(3xy(x + 5 - 7y)\) or is shown as \(3x\left(xy + 5y - 7y^2\right)\).