Question
Factor the GCF from the polynomial. \[ 18 x^{3} y^{2}-16 x^{3} y-10 x^{2} y \]
Ask by Schmidt Lang. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The factored form of the polynomial is \( 2x^{2} y (9xy - 8x - 5) \).
Solution
To factor out the **Greatest Common Factor (GCF)** from the polynomial
\[
18 x^{3} y^{2} - 16 x^{3} y - 10 x^{2} y,
\]
follow these steps:
1. **Identify the GCF of the coefficients:**
- **Coefficients:** 18, -16, -10
- The GCF of 18, 16, and 10 is **2**.
2. **Identify the GCF of the variables:**
- **For \( x \):** The exponents are 3, 3, and 2. The smallest exponent is **2**, so the GCF for \( x \) is \( x^2 \).
- **For \( y \):** The exponents are 2, 1, and 1. The smallest exponent is **1**, so the GCF for \( y \) is \( y \).
3. **Combine the GCF of coefficients and variables:**
- **Overall GCF:** \( 2x^2y \).
4. **Factor out the GCF from each term:**
\[
\begin{align*}
18x^3y^2 &= 2x^2y \cdot 9xy, \\
-16x^3y &= 2x^2y \cdot (-8x), \\
-10x^2y &= 2x^2y \cdot (-5).
\end{align*}
\]
5. **Write the factored form:**
\[
2x^2y (9xy - 8x - 5).
\]
**Final Answer:**
\[
2x^{2} y \left(9 x y - 8 x - 5\right)
\]
Reviewed and approved by the UpStudy tutoring team
Like
error msg
Explain 
Simplify this solution Extra Insights
To factor out the greatest common factor (GCF) from the polynomial \( 18 x^{3} y^{2} - 16 x^{3} y - 10 x^{2} y \), we first identify the GCF of the coefficients and the variables in each term. The coefficients 18, 16, and 10 have a GCF of 2. Each term contains at least \( x^{2} \) and \( y \), so we will factor out \( 2x^{2}y \). Now, we divide each term by \( 2x^{2}y \): 1. \( \frac{18 x^{3} y^{2}}{2 x^{2} y} = 9 x y \) 2. \( \frac{-16 x^{3} y}{2 x^{2} y} = -8 x \) 3. \( \frac{-10 x^{2} y}{2 x^{2} y} = -5 \) Putting it all together gives us: \[ 2 x^{2} y (9 x y - 8 x - 5) \] Thus, the factored form of the polynomial is: \[ 2 x^{2} y (9 xy - 8x - 5) \]
