Chapter 6

Perform each operation. $$ \left(4 u^{2}+z^{2}-3 u^{2} z^{2}\right)-\left(u^{3}+3 z^{2}-3 u^{2} z^{2}\right) $$

Let \(f(x)=\frac{2 x^{3}+x^{2}}{98 x+49} .\) For what values of \(x\) is \(f(x)=1 ?\)

As the cost of a purchase that is less than \(\$ 5\) increases, the amount of change received from a five-dollar bill decreases. Is this inverse variation? Explain.

a. \(\frac{x-1}{x+2}+\frac{x+2}{x-1}\) b. \(\frac{x-1}{x+2} \cdot \frac{x+2}{x-1}\)

Simplify each expression. $$ \frac{x^{32}-1}{x^{16}-1} $$

Let \(f(x)=\frac{x^{3}+4 x^{2}}{25 x+100} .\) For what values of \(x\) is \(f(x)=1 ?\)

You've probably heard of Murphy's first law: If anything can go wrong, it will. Another of Murphy's laws is: The chances of a piece of bread falling with the grapejelly side down varies directly with the cost of the carpet. Write one of your own witty sayings using the phrase varies directly.

a. \(\frac{n+2}{4}-\frac{8}{4 n+8}\) b. \(\frac{n+2}{4} \cdot \frac{8}{4 n+8}\)

Simplify each expression. $$ \frac{20 m^{2}\left(m^{2}-1\right)-47 m\left(1-m^{2}\right)+24\left(m^{2}-1\right)}{4 m^{2}-m-3} $$

Simplify each expression. $$ \frac{a^{6}-64}{\left(a^{2}+2 a+4\right)\left(a^{2}-2 a+4\right)} $$

The Amazon. In Brazil, when the Amazon River is at low stage, the rate of flow is about \(5 \mathrm{mph}\). Suppose that a river guide can canoe in still water at a rate of \(r\) mph. a. Complete the table to find rational expressions that represent the time it would take the guide to canoe 3 miles downriver and to canoe 3 miles upriver on the Amazon. $$\begin{array}{|l|c|c|c|}\hline & \text { Rate (mph) } & \text { Time (hr) } & \text { Distance (mI) } \\\\\hline \text { Downriver } & r+5 & & 3 \\\\\hline \text { Upriver } & r-5 & & 3 \\\\\hline\end{array}$$ b. Find the difference in the times for the trips upriver and downriver. Express the result as a single rational expression.

Simplify each expression. $$ \frac{(p+q)^{3}+64}{(p+q)^{2}-16} $$

Explain how to find the least common denominator of a set of rational expressions.

Graph each rational function. Show the vertical asymptote as a dashed line and label it. $$ f(x)=\frac{1}{x-1} $$

Add the rational expressions by expressing them in terms of a common denominator \(24 b^{3}\). (Note: This is not the LCD.) An extra step has to be performed to obtain the correct result because the lowest common denominator was not used. What was the step? $$\frac{r}{4 b^{2}}+\frac{s}{6 b}$$

Graph each rational function. Show the vertical asymptote as a dashed line and label it. $$ f(x)=\frac{1}{x+4} $$

Write some comments to the student who wrote the following solution, pointing out where she made an error. Subtract: $$\begin{aligned}\frac{1}{x}-\frac{x+1}{x} &=\frac{1-x+1}{x} \\\&=\frac{2-x}{x} \end{aligned}$$

Solve each equation. $$a(a-6)=-9$$

Solve each equation. $$x^{2}-\frac{1}{2}(x+1)=0$$

Solve each equation. $$y^{3}+y^{2}=0$$

Solve each equation. $$5 x^{2}=6-13 x$$

Find two rational expressions, each with denominator \(x^{2}+5 x+6,\) such that their sum is \(\frac{1}{x+2}\).

$$\text { Add: } x^{-1}+x^{-2}+x^{-3}+x^{-4}+x^{-5}$$

Perform the operations and simplify the result when possible. $$\left(\frac{3}{x-3}-\frac{1}{x}\right) \div \frac{12 x+18}{x^{3}-9 x}$$

Perform the operations and simplify the result when possible. $$\frac{13 x+39}{4 x^{2}+24 x+36} \div\left(\frac{7}{3 x+9}-\frac{5}{4 x+12}\right)$$

Perform the operations and simplify the result when possible. $$\left(\frac{3 x}{x+1}-\frac{6}{x^{2}-1}+\frac{4}{x-1}\right)\left(\frac{x^{3}-1}{9 x^{2}-4}\right)$$

Perform the operations and simplify the result when possible. $$\left(\frac{3}{a+2 b}-\frac{2 b}{a^{2}+2 a b}\right) \div\left(\frac{3 b}{a^{2}+2 a b}+\frac{5}{a}\right)$$