Chapter 11.3, Problem 4P

a.

To determine

To divide the expressions $\left(-7-10y+6y^2\right)÷\left(4+3y\right)$ .

Answer to Problem 4P

  $2y-6+\frac{17}{3y+4}$

Explanation of Solution

Given information:

The expression $\left(-7-10y+6y^2\right)÷\left(4+3y\right)$

Concept used:

Exponent properties: If a , b are real numbers and m , n are whole numbers, then

• Product Property: $a^m\cdot a^n=a^{m+n}$
• Quotient property: $\frac{a^m}{a^n}=a^{m-n}$
• Power Property: ${\left(a^m\right)}^n=a^{m\cdot n}$
• Product to a Power Property: ${\left(ab\right)}^m=a^m{b}^m$

Calculation:

In order to divide the given expression, first rewrite the dividend and divisor in standard form and then use long division to divide as shown below,

  $3y+4\begin{array}{c}\hfill 2y-6\\ \hfill \overline{)\begin{array}{l}6y^2-10y-7\\ \underset{\_}{-\left(6y^2+8y\right)}\\ -18y-7\\ \underset{\_}{-\left(-18y-24\right)}\\ 17\end{array}}\end{array}$

Thus,on dividing it gives $2y-6+\frac{17}{3y+4}$ .

b.

To determine

To divide the expressions $\left(21a+2+18a^2\right)÷\left(5+6a\right)$ .

Answer to Problem 4P

  $3a+1-\frac{3}{6a+5}$

Explanation of Solution

Given information:

The expression $\left(21a+2+18a^2\right)÷\left(5+6a\right)$

Concept used:

Exponent properties: If a , b are real numbers and m , n are whole numbers, then

• Product Property: $a^m\cdot a^n=a^{m+n}$
• Quotient property: $\frac{a^m}{a^n}=a^{m-n}$
• Power Property: ${\left(a^m\right)}^n=a^{m\cdot n}$
• Product to a Power Property: ${\left(ab\right)}^m=a^m{b}^m$

Calculation:

In order to divide the given expression, first rewrite the dividend and divisor in standard form and then use long division to divide as shown below,

  $6a+5\begin{array}{c}\hfill 3a+1\\ \hfill \overline{)\begin{array}{l}18a^2+21a+2\\ \underset{\_}{-\left(18a^2+15a\right)}\\ 6a+2\\ \underset{\_}{-\left(6a+5\right)}\\ -3\end{array}}\end{array}$

Thus, on dividing it gives $3a+1-\frac{3}{6a+5}$ .

c.

To determine

How to check the answer to above problems.

Explanation of Solution

Calculation:

In order to check the solutions of the above problems, first multiply the divisor with the quotient and then add the remainder. If you get the original dividend, then your answer is correct.