Chapter 6

Consider the first step of the division process for \(2 x ^ { 2 } - 1 \longdiv { 4 x ^ { 4 } + 0 x ^ { 3 } + 0 x ^ { 2 } + 0 x - 1 }\) How many times does \(2 x^{2}\) divide \(4 x^{4} ?\)

Complete each solution. $$ \begin{aligned} \frac{x^{2}-x-6}{4 x^{2}+16 x} \div \frac{x-3}{x+4} &=\frac{x^{2}-x-6}{4 x^{2}+16 x} \cdot \\ &=\frac{(x+2)(x+4)}{(x+4)(x-3)} \\ &=\frac{x+2}{\underline{\phantom{xx}}} \end{aligned} $$

Fill in the blanks. The rational function \(f(x)=\frac{9 x}{x-10}\) is __ for \(x=10\) In other words, there is __ a on the domain of the function: \(x \neq 10\)

Complete each solution to simplify the rational expression. a. Fill in the blank: The expression \(\frac{\frac{a}{b}}{\frac{c}{d}}\) is equivalent to \(\frac{a}{b} \square \frac{c}{d}\) b. What is the numerator and what is the denominator of the following complex fraction? $$ \frac{6-k-\frac{5}{k}}{k^{2}-9} $$

Fill in the blanks. Rather than substituting 8 for \(x\) in \(P(x)=6 x^{3}-x^{2}-17 x+9\) we can divide the polynomial _____ by ____ to find \(P(8)\).

Solve equation. \(\frac{1}{4}+\frac{9}{x}=1\)

Write \(\frac{41}{9}\) hours using a mixed number.

Consider the following two procedures. $$i \frac{x^{2}-2 x}{x^{2}+4 x-12}=\frac{x(x-2)}{(x+6)(x-2)}=\frac{x}{x+6}$$ $$\text { ii. } \frac{x}{x+6}=\frac{x}{x+6} \cdot \frac{x-2}{x-2}=\frac{x^{2}-2 x}{(x+6)(x-2)}$$ a. In which of these procedures are we building a rational expression? For what type of problem is this procedure often necessary? b. What name is used to describe the other procedure?

Complete each solution. A student checks her answers with those in the back of her textbook. Determine whether they are equivalent. $$ \begin{array}{|c|c|c|} \hline \text { Student's answer } & \text { Book's answer } & \text { Equivalent? } \\ \hline \frac{-x^{10}}{y^{2}} & -\frac{x^{10}}{y^{2}} & \\ \hline \frac{x-3}{x+3} & \frac{3-x}{x+3} & \\ \hline \frac{a+b}{(2-x)(c+d)} & -\frac{a+b}{(x-2)(c+d)} & \\ \hline \end{array} $$

A student checks her answers with those in the back of her textbook. Determine whether they are equivalent. $$ \begin{array}{|c|c|c|} \hline \text { Student's answer } & \text { Book's answer } & \text { Equivalent? } \\ \hline \frac{3+2 t}{t^{2}+2 t} & \frac{2 t+3}{t(t+2)} & \\ \hline \frac{5-3 x^{2}}{x+x^{2}} & -\frac{3 x^{2}-5}{x^{2}+x} & \\ \hline \frac{3 x y(y+x)}{(2 y-x)(2 y+3 x)} & \frac{3 x y^{2}+3 x^{2} y}{(2 y+x)(2 y-3 x)} & \\ \hline \end{array} $$

Fill in the blanks. For \(P(x)=x^{3}-4 x^{2}+x+6,\) suppose we know that \(P(3)=0\) Then _____ is a factor of \(x^{3}-4 x^{2}+x+6\).

Solve equation. \(\frac{1}{3}-\frac{10}{x}=-3\)

The LCD for \(\frac{2 x+1}{x^{2}+5 x+6}\) and \(\frac{3 x}{x^{2}-4}\) is $$\mathrm{LCD}=(x+2)(x+3)(x-2)$$ If we want to subtract these rational expressions, what form of 1 should be used: a. to build \(\frac{2 x+1}{x^{2}+5 x+6} ?\) b. to build \(\frac{3 x}{x^{2}-4} ?\)

Complete each solution. a. Write \(5 x^{2}+35 x\) as a fraction. b. What is the reciprocal of \(5 x^{2}+35 x ?\)

Fill in the blanks to simplify \(\frac{x-y}{y-x}\) $$ \frac{x-y}{y-x}=\frac{-y+}{y-x}=\frac{(y-x)}{(y-x)}= $$

Complete each synthetic division. Divide \(6 x^{3}+x^{2}-23 x+2\) by \(x-2\)

Tell whether each relationship suggests direct or inverse variation. Recycling. The amount of money you receive and the number of aluminum cans you return

Solve equation. \(\frac{1}{a}=\frac{1}{3}-\frac{2}{3 a}\)

Roofing. \(\quad\) A homeowner estimates that it will take him 7 days to roof his house. A professional roofer estimates that he could roof the house in 4 days. How long will it take if the homeowner helps the roofer?

Consider the following factorizations of the denominators of two rational expressions: $$(x-2)(x-2) \text { and } 3(x-2)$$ a. What is the greatest number of times the factor 3 appears in any one factorization? b. What is the greatest number of times the factor \(x-2\) appears in any one factorization? c. What is the LCD of the rational expressions?

If a polynomial is divided by \(3 a-2\) and the quotient is \(3 a^{2}+5\) with a remainder of \(6,\) how do we write the result?

Multiply, and then simplify, if possible. See Objective 1. $$ \frac{3}{4} \cdot \frac{11}{3} $$

Simplify each expression. a. \(\frac{3 \cdot 5 \cdot x \cdot y \cdot y}{5 \cdot 7 \cdot x \cdot x \cdot x \cdot y}\) b. \(\frac{(x+8)(x-3)}{(x+2)(x+8)}\) c. \(\frac{a^{3}(a-9)}{(9-a)(9+a)}\)

Simplify each complex fraction. See Examples 1 and 2. $$ \frac{\frac{3 b^{7}}{4}}{\frac{b^{9}}{2}} $$

Complete each synthetic division. Divide \(2 x^{3}-4 x^{2}-25 x+15\) by \(x+3\)

Tell whether each relationship suggests direct or inverse variation. Karate. The force needed to break a board and the length of the board

Solve equation. \(\frac{1}{b}=\frac{1}{8}-\frac{3}{8 b}\)

Decorating. One crew can put up holiday decorations in a department store in 12 hours. A second crew can put up the decorations in 15 hours. How long will it take if both crews work together to decorate the store?

The factorizations of the denominators of two rational expressions follow. Find the LCD. $$\left.\begin{array}{l}2 \cdot 3 \cdot a \cdot a \cdot a \\\2 \cdot 3 \cdot 3 \cdot a \cdot a \end{array}\right\\} \mathrm{LCD}=\square$$

A polynomial is divided by \(3 a-2 .\) The quotient is \(3 a^{2}+5\) with a remainder of \(-6 .\) Write the answer to the division in two ways.

Multiply, and then simplify, if possible. See Objective 1. $$ \frac{13}{6} \cdot \frac{6}{21} $$

Simplify each rational expression, if possible. a. \(\frac{x+8}{x}\) b. \(\frac{3 a^{2}+23}{a^{2}}\)

Tell whether each relationship suggests direct or inverse variation. Tools. The force you must exert on the handle of a wrench to loosen a bolt and the length of the handle

Solve equation. \(\frac{18}{y+1}+\frac{2}{5}=4\)

Factor each denominator completely. a. \(\frac{17}{40 x^{2}}\) b. \(\frac{x+25}{2 x^{2}-6 x}\) c. \(\frac{n^{2}+3 n-4}{n^{2}-64}\)

List three ways we can use symbols to write \(x^{2}-x-12\) divided by \(x-4 .\)

Multiply, and then simplify, if possible. See Objective 1. $$ \frac{15}{24} \cdot \frac{16}{25} $$

Use synthetic division to perform each division. See Example 1. $$ \left(4 x^{2}-5 x-6\right) \div(x-2) $$

Solve equation. \(\frac{2}{3}+\frac{10}{a+2}=4\)

Tell whether each relationship suggests direct or inverse variation. Anatomy. The volume of blood pumped from your heart each minute and your pulse rate

Groundskeeping. It takes a groundskeeper 45 minutes to prepare a Little League baseball field for a game. It takes his assistant 55 minutes to prepare the same field. How long will it take if they work together to prepare the field?

By what must \(y-4\) be multiplied to obtain \(4-y ?\)

Is the following statement true or false? Justify your answer. $$ 2 x^{3}-9=2 x^{3}+0 x^{2}+0 x-9 $$

Multiply, and then simplify, if possible. See Objective 1. $$ \frac{49}{36} \cdot \frac{18}{35} $$

For what value(s) of \(x\) is each function undefined? a. \(f(x)=\frac{x-7}{x}\) b. \(\quad f(x)=\frac{x+1}{x-3}\) c. \(f(x)=\frac{x^{2}-2}{x(x+8)}\) d. \(f(x)=\frac{8 x}{(x-1)(x+1)}\)

Use synthetic division to perform each division. See Example 1. $$ \left(5 x^{2}-27 x+10\right) \div(x-5) $$

Solve equation. \(\frac{1}{2}+\frac{x}{x-1}=3\)

Farming. In 10 minutes, a conveyor belt can move \(1,000\) bushels of corn into the storage bin shown. A smaller belt can move \(1,000\) bushels to the storage bin in 14 minutes. If both belts are used, how long will it take to move \(1,000\) bushels to the storage bin?

Complete each solution. $$\frac{6 x-1}{3 x-1}+\frac{3 x-2}{3 x-1}=\frac{6 x-1+\square }{3 x-1} $$ $$=\frac{9 x-\square}{3 x-1}$$ $$=\frac{3(\quad)}{3 x-1}$$ $$=\square$$

Simplify. Write answers using positive exponents. \(\frac{4 x^{2} y^{3}}{8 x^{5} y^{2}}\)