Q25. - Algebra 1

Question

Use the Distributive Property to factor each polynomial.

$24am-9an+40bm-15bn$

Step-by-Step Solution

Answer

The factorization of the given polynomial is $\left(8m-3n\right)\left(3a+5b\right)$.

Step 1. Find the greatest common factor of 24 a m and 9 a n .

Find the factorization of $24am$ and $9an$.

$$\begin{gathered} 24am=2\cdot 2\cdot 2\cdot 3\cdot a\cdot m \\ 9an=3\cdot 3\cdot a\cdot n \end{gathered}$$

From the factorization of $24am$ and $9an$, it can be noticed that the greatest common factor of $24am$ and $9an$ is $3a$.

Therefore, the greatest common factor of $24am$ and $9an$ is $3a$.

Step 2. Find the greatest common factor of 40 b m and 15 b n .

Find the factorization of $40bm$ and $15bn$.

$$\begin{gathered} 40bm=2\cdot 2\cdot 2\cdot 5\cdot b\cdot m \\ 15bn=3\cdot 5\cdot b\cdot n \end{gathered}$$

From the factorization of $40bm$ and $15bn$, it can be noticed that the greatest common factor of $40bm$ and $15bn$ is $5b$.

Therefore, the greatest common factor of $40bm$ and $15bn$ is $5b$.

 

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

Therefore, it is obtained that:

$24am-9an+40bm-15bn=3a\left(8m\right)-3a\left(3n\right)+5b\left(8m\right)-5b\left(3n\right)$

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:

$24am-9an+40bm-15bn=3a\left(8m\right)-3a\left(3n\right)+5b\left(8m\right)-5b\left(3n\right)$

$$\begin{gathered} =3a\left(8m-3n\right)+5b\left(8m-3n\right) \\ =\left(8m-3n\right)\left(3a+5b\right)\left(\text{take out }8m-3n\text{ as common factor}\right) \end{gathered}$$

Therefore, the factorization of the given polynomial is $\left(8m-3n\right)\left(3a+5b\right)$.