Chapter 11.3, Problem 1P

a.

To determine

To divide the expressions $\left(4a^3+10a^2+3a\right)÷2a^2$ .

Answer to Problem 1P

  $2a+5+\frac{3}{2a}$

Explanation of Solution

Given information:

The expression $\left(4a^3+10a^2+3a\right)÷2a^2$

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 it in fractional form and then separate the terms. Then simplify further using the quotient property of exponents as shown below,

  $\begin{array}{c}\left(4a^3+10a^2+3a\right)÷2a^2=\frac{4a^3+10a^2+3a}{2a^2}\\ =\frac{4a^3}{2a^2}+\frac{10a^2}{2a^2}+\frac{3a}{2a^2}\\ =2a+5+\frac{3}{2a}\end{array}$

Thus, the quotient of the given expressions is $2a+5+\frac{3}{2a}$ .

b.

To determine

To divide the expressions $\left(5b^4-15b^2+1\right)÷5b^3$ .

Answer to Problem 1P

  $b-\frac{3}{b}+\frac{1}{5b^3}$

Explanation of Solution

Given information:

The expression $\left(5b^4-15b^2+1\right)÷5b^3$

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 it in fractional form and then separate the terms. Then simplify further using the quotient property of exponents as shown below,

  $\begin{array}{c}\left(5b^4-15b^2+1\right)÷5b^3=\frac{5b^4-15b^2+1}{5b^3}\\ =\frac{5b^4}{5b^3}-\frac{15b^2}{5b^3}+\frac{1}{5b^3}\\ =b-\frac{3}{b}+\frac{1}{5b^3}\end{array}$

Thus, the quotient of the given expressions is $b-\frac{3}{b}+\frac{1}{5b^3}$ .

c.

To determine

To divide the expressions $\left(12c^4+18c^2+9c\right)÷6c$ .

Answer to Problem 1P

  $2c^3+3c+\frac{3}{2}$

Explanation of Solution

Given information:

The expression $\left(12c^4+18c^2+9c\right)÷6c$

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 it in fractional form and then separate the terms. Then simplify further using the quotient property of exponents as shown below,

  $\begin{array}{c}\left(12c^4+18c^2+9c\right)÷6c=\frac{12c^4+18c^2+9c}{6c}\\ =\frac{12c^4}{6c}+\frac{18c^2}{6c}+\frac{9c}{6c}\\ =2c^3+3c+\frac{3}{2}\end{array}$

Thus, the quotient of the given expressions is $2c^3+3c+\frac{3}{2}$ .