Chapter 1.4, Problem 5BGP

To determine

The expression for the verbal expression “5 times the difference of $q$ squared and $r$ plus 8 times the sum of $3q$ and $2r$ ”, simplify it and name the properties used.

Answer to Problem 5BGP

The simplified expression for the verbal expression “5 times the difference of $q$ squared and $r$ plus 8 times the sum of $3q$ and $2r$ ” is $5q^2+24q+11r$ .

Explanation of Solution

Given:

The verbal expression is 5 times the difference of $q$ squared and $r$ plus 8 times the sum of $3q$ and $2r$ .

Calculation:

The verbal expression “the difference of $q$ squared and $r$ ” implies that there is a subtraction of $q^2$ and r, i.e., $q^2-r$ .

Now, 5 times of $q^2-r$ is referred as $5\left(q^2-r\right)$ .

The verbal expression “sum of $3q$ and $2r$ ” implies that there is addition of two expressions, i.e., $3q+2r$ .

8 times $3q+2r$ is expressed as $8\left(3q+2r\right)$ . The sum of $5\left(q^2-r\right)$ and $8\left(3q+2r\right)$ is expressed as $5\left(q^2-r\right)+8\left(3q+2r\right)$ .

Simplify the expression $5\left(q^2-r\right)+8\left(3q+2r\right)$ as,

  $\begin{array}{c}5\left(q^2-r\right)+8\left(3q+2r\right)=5q^2-5r+24q+16r\left(\text{Distributive Property}\right)\\ =5q^2+24q+16r-5r\left(\text{Commutative Property}\right)\\ =5q^2+24q+11r\end{array}$

Therefore, the simplified expression for “5 times the difference of $q$ squared and $r$ plus 8 times the sum of $3q$ and $2r$ ” is $5q^2+24q+11r$ .