Question

Use the Distributive Property to factor each polynomial.

$a^2-4ac+ab-4bc$

Short answer

The factorization of the given polynomial is $\left(a-4c\right)\left(a+b\right)$.

Step-by-step solution

Step 1. Find the greatest common factor of a2 and 4ac.

Find the factorization of $a^2$and $4ac$.

$$\begin{gathered} a^2=a\cdot a \\ 4ac=2\cdot 2\cdot a\cdot c \end{gathered}$$

From the factorization of $a^2$and $4ac$, it can be noticed that the greatest common factor of $a^2$and $4ac$is $a$.

Therefore, the greatest common factor of $a^2$and $4ac$is $a$.

Step 2. Find the greatest common factor of ab and 4bc.

Find the factorization of $ab$ and $4bc$.

$$\begin{gathered} ab=a\cdot b \\ 4bc=2\cdot 2\cdot b\cdot c \end{gathered}$$

From the factorization of $ab$and $4bc$, it can be noticed that the greatest common factor of $ab$and $4bc$is $b$.

Therefore, the greatest common factor of $ab$and $4bc$is $b$.

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

Therefore, it is obtained that:

$a^2-4ac+ab-4bc=a\left(a\right)-a\left(4c\right)+b\left(a\right)-b\left(4c\right)$

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="1647949311355" $$\begin{gathered} a^2-4ac+ab-4bc=a\left(a\right)-a\left(4c\right)+b\left(a\right)-b\left(4c\right) \\ =a\left(a-4c\right)+b\left(a-4c\right) \\ =\left(a-4c\right)\left(a+b\right)\left(\text{takeout}a-4c\text{ascommonfactor}\right) \end{gathered}$$

Therefore, the factorization of the given polynomial is $\left(a-4c\right)\left(a+b\right)$.