Question

Use the Distributive Property to factor each polynomial.

$2x^2-3xz-2xy+3yz$

Short answer

The factorization of the given polynomial is $\left(2x-3z\right)\left(x-y\right)$.

Step-by-step solution

Step 1. Find the greatest common factor of 2x2 and 3xz.

Find the factorization of $2x^2$ and $3xz$.

$$\begin{gathered} 2x^2=2\cdot x\cdot x \\ 3xz=3\cdot x\cdot z \end{gathered}$$

From the factorization of $2x^2$and $3xz$, it can be noticed that the greatest common factor of $2x^2$and $3xz$is $x$.

Therefore, the greatest common factor of $2x^2$and $3xz$is $x$.

Step 2. Find the greatest common factor of 2xy and 3yz.

Find the factorization of $2xy$ and $3yz$.

$$\begin{gathered} 2xy=2\cdot x\cdot y \\ 3yz=3\cdot y\cdot z \end{gathered}$$

From the factorization of $2xy$and $3yz$, it can be noticed that the greatest common factor of $2xy$and $3yz$is $y$.

Therefore, the greatest common factor of $2xy$and $3yz$is $y$.

Step 3. Write each term as the product of the greatest common factor and the remaining factors.

Therefore, it is obtained that:

$2x^2-3xz-2xy+3yz=x\left(2x\right)-x\left(3z\right)-y\left(2x\right)+y\left(3z\right)$

Step 4. 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:

role="math" localid="1647949844013" $$\begin{gathered} 2x^2-3xz-2xy+3yz=x\left(2x\right)-x\left(3z\right)-y\left(2x\right)+y\left(3z\right) \\ =x\left(2x-3z\right)-y\left(2x-3z\right) \\ =\left(2x-3z\right)\left(x-y\right)\left(\text{takeout}2x-3z\text{ascommonfactor}\right) \end{gathered}$$

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