Question

Use the Distributive Property to factor each polynomial.

$14x^{2}y-21xy+35x{y}^2$

Short answer

The factorization of the given polynomial is $\mathbf{7}\mathit{x}\mathit{y}\left(2x-3+5y\right)$.

Step-by-step solution

Step 1. Find the greatest common factor of 14x2y,21xy and 35xy2.

Find the factorization of $14x^{2}y,21xy$and $35x{y}^2$.

$$\begin{gathered} 14x^{2}y=2\cdot 7\cdot x\cdot x\cdot y \\ 21xy=3\cdot 7\cdot x\cdot y \\ 35x{y}^2=5\cdot 7\cdot x\cdot y\cdot y \end{gathered}$$

From the factorization of $14x^{2}y,21xy$and $35x{y}^2$, it can be noticed that the greatest common factor of $14x^{2}y,21xy$and $35x{y}^2$is$7\cdot x\cdot y$ or $7xy$.

Therefore, the greatest common factor of $14x^{2}y,21xy$and $35x{y}^2$is $7xy$.

Step 2. Write each term as the product of the greatest common factor and the remaining factors.

Therefore, it is obtained that:

$14x^{2}y-21xy+35x{y}^2=7xy\left(2x\right)-7xy\left(3\right)+7xy\left(5y\right)$

Step 3. Use the distributive property to factor out the greatest common factor.

The distributive property states that:

$ab+ac=a\left(b+c\right)$

Now, use the distributive property to factor out the greatest common factor.

Therefore, it is obtained that:

$$\begin{gathered} 14x^{2}y-21xy+35x{y}^2=7xy\left(2x\right)-7xy\left(3\right)+7xy\left(5y\right) \\ =7xy\left(2x-3+5y\right) \end{gathered}$$

Therefore, the factorization of the given polynomial is $7xy\left(2x-3+5y\right)$.