Chapter 5.10, Problem 2WE

To determine

To factor the below expression:

  $2\left(r-s\right)+t\left(r-s\right)$

And to check by multiplying the factors.

Answer to Problem 2WE

  $2\left(r-s\right)+t\left(r-s\right)=\left(r-s\right)\left(2+t\right)$

Explanation of Solution

Given:

  $2\left(r-s\right)+t\left(r-s\right)$

Concept Used:

Distributive property: $ab\pm ac=a\left(b\pm c\right)$

Calculation:

To factor the expression:

  $2\left(r-s\right)+t\left(r-s\right)$

Use the distributive property, $ab\pm ac=a\left(b\pm c\right)$ .

Here, in given expression note that $\left(r-s\right)$ is common in each term factor it out using distributive property, as

  $2\left(r-s\right)+t\left(r-s\right)=\left(r-s\right)\left(2+t\right)$

Thus, factor of given expression is:

  $2\left(r-s\right)+t\left(r-s\right)=\left(r-s\right)\left(2+t\right)$

Now, to check the multiply these factors and see if it gives the original expression.

For that again use distributive property $a\left(b\pm c\right)=ab\pm ac$ , as

  $\begin{array}{c}\left(r-s\right)\left(2+t\right)=\left(2+t\right)\left(r-s\right)\\ =2\left(r-s\right)+t\left(r-s\right)\\ =\text{Original expression}\text{.}\end{array}$

Thus, the original expression can be achieved by multiplying the factors of the expression.