Chapter SH, Problem 14.6.6EP

To determine

To divide: the given expression

Answer to Problem 14.6.6EP

The required result is $n+6m+3$

Explanation of Solution

Given:

  $\left(2mn+12m^2+6m\right)÷\left(2m\right)$

Calculation:

Essential understanding:

To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.

  $\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}$ .

Write the rational expression,

  $\left(2mn+12m^2+6m\right)÷\left(2m\right)=\frac{2mn+12m^2+6m}{2m}$

Divide each term by 2m .

  $\begin{array}{l}\frac{2mn+12m^2+6m}{2m}=\frac{2mn}{2m}+\frac{12m^2}{2m}+\frac{6m}{2m}\\ =\frac{2}{2}\cdot \frac{m}{m}\cdot \frac{n}{1}\text{+}\frac{12}{2}\cdot \frac{m^2}{m}\text{+}\frac{6}{2}\cdot \frac{m}{m}\text{ (associative property)}\\ \text{ =1}\cdot 1\cdot n+6\cdot m+3\cdot 1\text{ (quotient of powers)}\\ =n+6m+3\text{ (identity)}\end{array}$

Conclusion:Hence, the result is $n+6m+3$ .