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Intermediate Algebra: Functions and Graphs

Katherine Yoshiwara

Section 8.6 Chapter 8 Summary and Review

Subsection 8.6.1 Glossary

  • polynomial
  • degree
  • descending powers
  • ascending powers
  • lead coefficient
  • algebraic fraction
  • vertical asymptote
  • horizontal asymptote
  • rational function
  • lowest common denominator
  • building factor
  • like fractions
  • common factor
  • complex fraction
  • proportion
  • extraneous solution

Subsection 8.6.2 Key Concepts

  1. Polynomial Function.

    A polynomial function has the form
    \begin{equation*} f(x) = a_n x^n + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} + \cdots + a_2x^2 + a_1x + a_0 \end{equation*}
    where \(a_0\text{,}\) \(a_1\text{,}\) \(a_2\text{,}\) \(\ldots\text{,}\) \(a_n\) are constants and \(a_n \ne 0\text{.}\) The coefficient \(a_n\) of the highest power term is called the lead coefficient.

  2. Degree of a Product.

    The degree of a product of non-zero polynomials is the sum of the degrees of the factors. That is:
    If \(P(x)\) has degree \(m\) and \(Q(x)\) has degree \(n\text{,}\) then their product \(P(x)Q(x)\) has degree \(m+n\text{.}\)

  3. Special Products of Binomials.

    \begin{align*} \amp(a + b)^2 = (a + b) (a + b) = a^2 + 2ab + b^2\\ \amp(a - b)^2 = (a - b) (a - b) = a^2 - 2ab + b^2\\ \amp(a + b) (a - b)= a^2 -b^2 \end{align*}

  4. Cube of a Binomial.

    1. \(\displaystyle (x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3\)
    2. \(\displaystyle (x - y)^3 = x^3 - 3x^2y + 3xy^2 - y^3\)

  5. Factoring the Sum or Difference of Two Cubes.

    1. \(\displaystyle x^3 + y^3 = (x + y)(x^2 - xy + y^2)\)
    2. \(\displaystyle x^3 - y^3 = (x - y)(x^2 + xy + y^2)\)

  6. Fundamental Principle of Fractions.

    We can multiply or divide the numerator and denominator of a fraction by the same nonzero factor, and the new fraction will be equivalent to the old one.
    \begin{equation*} \dfrac{a \cdot c}{b \cdot c} = \dfrac{a}{b}~~~~~~\text{if}~~~~~~b,c \ne 0 \end{equation*}
  7. When we cancel common factors, we are dividing. Because division is the inverse or opposite operation for multiplication, we can cancel common factors, but we cannot cancel common terms.

  8. To reduce an algebraic fraction.

    1. Factor the numerator and the denominator.
    2. Divide the numerator and denominator by any common factors.

  9. Rational Function.

    A rational function has the form
    \begin{equation*} f(x) = \dfrac{P(x)}{Q(x)} \end{equation*}
    where \(P(x)\) and \(Q(x)\) are polynomials. A rational function is undefined at any \(x\)-values where \(Q(x) = 0\text{.}\)

  10. Operations on Fractions.

    • If \(b,~d \ne 0\text{,}\) then
      \begin{equation*} \dfrac{a}{b} \cdot \dfrac{c}{d} = \dfrac{ac}{bd} \end{equation*}
    • If \(b,~c,~d \ne 0\text{,}\) then
      \begin{equation*} \dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \cdot \dfrac{d}{c} \end{equation*}
    • If \(c \ne 0\text{,}\) then
      \begin{equation*} \dfrac{a}{b} + \dfrac{c}{d} = \dfrac{a+b}{c} \end{equation*}
      \begin{equation*} \dfrac{a}{b} - \dfrac{c}{d} = \dfrac{a-b}{c} \end{equation*}

  11. To multiply algebraic fractions:.

    1. Factor each numerator and denominator.
    2. If any factor appears in both a numerator and a denominator, divide out that factor.
    3. Multiply the remaining factors of the numerator and the remaining factors of the denominator.
    4. Reduce the product if necessary.

  12. To divide algebraic fractions:.

    1. Take the reciprocal of the second fraction and change the operation to multiplication.
    2. Follow the rules for multiplication of fractions.

  13. To add or subtract algebraic fractions.

    1. Find the lowest common denominator (LCD) for the fractions.
    2. Build each fraction to an equivalent one with the same denominator.
    3. Add or subtract the resulting like fractions: Add or subtract their numerators, and keep the same denominator.
    4. Reduce the sum or difference if necessary.

  14. To Find the LCD.

    1. Factor each denominator completely.
    2. Include each different factor in the LCD as many times as it occurs in any one of the given denominators.

  15. To simplify a complex fraction.

    1. Find the LCD of all the fractions contained in the complex fraction.
    2. Multiply the numerator and the denominator of the complex fraction by the LCD.
    3. Reduce the resulting simple fraction, if possible.
  16. An algebraic fractrion is "improper" if the degree of the numerator is greater than the degree of the denominator. If it cannot be reduced, we can simplify the expression by treating it as a division of polynomials. The quotient will be the sum of a polynomial and a simpler algebraic fraction.
  17. If an equation contains more than one fraction, we can clear all the denominators at once by multiplying both sides by the LCD of the fractions.

  18. Property of Proportions.

    \begin{equation*} \blert{\text{If}~~\dfrac{a}{b}=\dfrac{c}{d},~~~\text{then}~~~ad=bc,~~\text{as long as}~ b,d \ne 0} \end{equation*}
  19. Whenever we multiply an equation by an expression containing the variable, we should check that the solution obtained does not cause any of the fractions to be undefined.

Exercises 8.6.3 Chapter 8 Review Problems

Exercise Group.

For Problems 1 and 2, multiply.
1.
\((2x-5)(x^2-3x+2)\)
Answer.
\(2x^3-11x^2+19x-10\)
2.
\((b^2-2b-3)(2b^2+b-5)\)

Exercise Group.

For Problems 3 and 4, factor.
3.
\(8x^3-27z^3\)
Answer.
\((2x-3z)(4x^2+6xz+9z^2)\)
4.
\(1+125a^3b^3\)

5.

The expression \(\dfrac{n}{6 }(n - 1)(n - 2)\) gives the number of different 3-item pizzas that can be created from a list of \(n\) toppings.
  1. Write the expression as a polynomial.
  2. If Mitch’s Pizza offers 12 different toppings, how many different combinations for 3-item pizzas can be made?
  3. Use a table or graph to determine how many different toppings are needed in order to be able to have more than 1000 possible combinations for 3-item pizzas.
Answer.
  1. \(\displaystyle \dfrac{1}{6}n^3-\dfrac{1}{2}n^2+\dfrac{1}{3}n \)
  2. \(\displaystyle 220\)
  3. \(\displaystyle 20\)

6.

The expression \(n(n - 1)(n - 2)\) gives the number of different triple-scoop ice cream cones that can be created from a list of \(n\) flavors.
  1. Write the expression as a polynomial.
  2. If Zanner’s Ice Cream Parlor offers 21 flavors, how many different triple-scoop ice cream cones can be made?
  3. Use a table or graph to determine how many different flavors are needed in order to be able to have more than 10,000 possible triple-scoop ice cream cones.

7.

The radius, \(r\text{,}\) of a cylindrical can should be one-half its height, \(h\text{.}\)
  1. Express the volume, \(V\text{,}\) of the can as a function of its height.
  2. What is the volume of the can if its height is 2 centimeters? 4 centimeters?
  3. Graph the volume as a function of the height and verify your results of part (b) graphically. What is the approximate height of the can if its volume is 100 cubic centimeters?
Answer.
  1. \(\displaystyle V=\dfrac{\pi h^3}{4} \)
  2. \(2\pi\text{ cm}^3 \approx 6.28\text{ cm}^3 \text{;}\) \(16\pi\text{ cm}^3 \approx 50.27\text{ cm}^3 \)
  3. cubic

8.

The Twisty-Freez machine dispenses soft ice cream in a cone-shaped peak with a height 3 times the radius of its base. The ice cream comes in a round bowl with base diameter \(d\text{.}\)
  1. Express the volume, \(V\text{,}\) of Twisty-Freez in the bowl as a function of \(d\text{.}\)
  2. How much Twisty-Freez comes in a 3-inch diameter dish? A 4-inch dish?
  3. Graph the volume as a function of the diameter and verify your results of part (b) graphically. What is the approximate diameter of a Twisty-Freez if its volume is 5 cubic inches?

9.

A new health club opened up, and the manager kept track of the number of active members over its first few months of operation. The equation below gives the number, \(N\text{,}\) of active members, in hundreds, \(t\) months after the club opened.
\begin{equation*} N=\frac{44t}{40+t^2} \end{equation*}
  1. Use your calculator to graph the function \(N\) in the window
    \begin{align*} \text{Xmin} \amp = 0 \amp\amp \text{Xmax} = 20\\ \text{Ymin} \amp = 0 \amp\amp \text{Ymax} = 4 \end{align*}
  2. How many active members did the club have after 8 months?
  3. In which months did the club have 200 active members?
  4. When does the health club have the largest number of active members? What happens to the number of active members as time goes on?
Answer.
  1. rational function
  2. 338
  3. Months 2 and 02
  4. During month 6. The number of members eventually decreases to zero.

10.

A small lake in a state park has become polluted by runoff from a factory upstream. The cost for removing \(p\) percent of the pollution from the lake is given, in thousands of dollars, by
\begin{equation*} C=\frac{25p}{100-p} \end{equation*}
  1. Use your calculator to graph the function \(C\) on a suitable domain.
  2. How much will it cost to remove 40% of the pollution?
  3. How much of the pollution can be removed for $100,000 ?
  4. What happens to the cost as the amount of pollution to be removed increases? How much will it cost to remove all the pollution?

11.

The Explorer’s Club is planning a canoe trip to travel 90 miles up the Lazy River and return in 4 days. Club members plan to paddle for \(6\) hours each day, and they know that the current in the Lazy River is 2 miles per hour.
  1. Express the time it will take for the upstream journey as a function of their paddling speed in still water.
  2. Express the time it will take for the downstream journey as a function of their paddling speed in still water.
  3. Graph the sum of the two functions in the window
    \begin{align*} \text{Xmin} \amp = 0 \amp\amp \text{Xmax} = 18.8\\ \text{Ymin} \amp = 0 \amp\amp \text{Ymax} = 50 \end{align*}
    and find the point on the graph with \(y\)-coordinate 24. Interpret the coordinates of the point in the context of the problem.
  4. The Explorer’s Club would like to know what average paddling speed members must maintain in order to complete their trip in 4 days. Write an equation to describe this situation.
  5. Solve your equation to find the required paddling speed.
Answer.
  1. \(\displaystyle t_1=\dfrac{90}{v-2} \)
  2. \(\displaystyle t_2=\dfrac{90}{v+2} \)
  3. curve
  4. \(\displaystyle \dfrac{90}{v-2}+\dfrac{90}{v+2}=24 \)
  5. 8 mph

12.

Pam lives on the banks of the Cedar River and makes frequent trips in her outboard motorboat. The boat travels at 20 miles per hour in still water.
  1. Express the time it takes Pam to travel 8 miles upstream to the gas station as a function of the speed of the current.
  2. Express the time it takes Pam to travel 12 miles downstream to Marie’s house as a function of the speed of the current.
  3. Graph the two functions in the window
    \begin{align*} \text{Xmin} \amp = 0 \amp\amp \text{Xmax} = 10\\ \text{Ymin} \amp = 0 \amp\amp \text{Ymax} = 1 \end{align*}
    and find the coordinates of the intersection point. Interpret those coordinates in the context of the problem.
  4. Pam traveled to the gas station in the same time it took her to travel to Marie’s house. Write an equation to describe this situation.
  5. Solve your equation to find the speed of the current in the Cedar River.

Exercise Group.

For Problems 13–18, reduce the fraction to lowest terms.
13.
\(\dfrac{2a^2(a-1)^2}{4a(a-1)^3}\)
Answer.
\(\dfrac{a}{2(a-1)}\)
14.
\(\dfrac{4y-6}{6}\)
15.
\(\dfrac{2x^2y^3-4x^3y}{4x^2y}\)
Answer.
\(\dfrac{y^2-2x}{2}\)
16.
\(\dfrac{(x-2y)^2}{4y^2-x^2}\)
17.
\(\dfrac{a^2-6a+9}{2a^2-18}\)
Answer.
\(\dfrac{a-3}{2a+6)}\)
18.
\(\dfrac{4x^2y^2+4xy+1}{4x^2y^2-1}\)

Exercise Group.

For Problems 19–26, write the expression as a single fraction in lowest terms.
19.
\(\dfrac{2a^2}{3b} \cdot \dfrac{15b^2}{a}\)
Answer.
\(10ab\)
20.
\(\dfrac{-1}{3} ab^2\cdot \dfrac{3}{4}a^3b\)
21.
\(\dfrac{4x+6}{2x} \cdot \dfrac{6x^2}{(2x+3)^2}\)
Answer.
\(\dfrac{6x}{2x+3}\)
22.
\(\dfrac{4x^2-9}{3x-3} \cdot \dfrac{x^2-1}{4x-6}\)
23.
\(\dfrac{a^2-a-2}{a^2-4} \div \dfrac{a^2+2a+1}{a^2-2a}\)
Answer.
\(\dfrac{a^2-2a}{a^2+3a+2}\)
24.
\(\dfrac{a^3-8b^3}{a^2b} \div \dfrac{a^2-4ab+4b^2}{ab^2}\)
25.
\(1 \div \dfrac{4x^2-1}{2x+1}\)
Answer.
\(\dfrac{1}{2x-1}\)
26.
\(\dfrac{y^2+2y}{3x} \div (4y)\)

Exercise Group.

For Problems 27–30, divide.
27.
\(\dfrac{36x^6-28x^4+16x^2-4}{4x^4}\)
Answer.
\(9x^2-7+\dfrac{4}{x^2}-\dfrac{1}{x^4} \)
28.
\(\dfrac{y^3+3y^2-2y-4}{y+1}\)
29.
\(\dfrac{x^3-4x^2+2x+3}{x-2}\)
Answer.
\(x^2-2x-2-\dfrac{1}{x-2}\)
30.
\(\dfrac{x^2+2x^3-1}{2x-1}\)

Exercise Group.

For Problems 31–36, write the expression as a single fraction in lowest terms.
31.
\(\dfrac{x+2}{3x} - \dfrac{x-4}{3x}\)
Answer.
\(\dfrac{2}{x}\)
32.
\(\dfrac{5}{6}b - \dfrac{1}{3}b + \dfrac{3}{4}b\)
33.
\(\dfrac{3}{2x-6} - \dfrac{4}{x^2-9}\)
Answer.
\(\dfrac{3x+1}{2(x-3)(x+3)}\)
34.
\(\dfrac{1}{y^2+4y+4} + \dfrac{3}{y^2-4}\)
35.
\(\dfrac{2a+1}{a-3} - \dfrac{-2}{a^2-4a+3}\)
Answer.
\(\dfrac{2a^2-a+1}{(a-3)(a-1{})}\)
36.
\(a - \dfrac{1}{a^2+2a+1} + \dfrac{3}{a^2-1}\)

Exercise Group.

For Problems 37–40, write the complex fraction as a simple fraction in lowest terms.
37.
\(\dfrac{\dfrac{3}{4} - \dfrac{1}{2}}{\dfrac{3}{4} + \dfrac{1}{2}}\)
Answer.
\(\dfrac{1}{5}\)
38.
\(\dfrac{y-\dfrac{2y}{x}}{1+\dfrac{2}{x}}\)
39.
\(\dfrac{x-4}{x-\dfrac{16}{x}}\)
Answer.
\(\dfrac{x}{x+4} \)
40.
\(\dfrac{\dfrac{1}{x-1}}{1-\dfrac{1}{x^2}} \)

Exercise Group.

For Problems 41–44, solve.
41.
\(\dfrac{y+3}{y+5}=\dfrac{1}{3}\)
Answer.
\(-2\)
42.
\(\dfrac{z^2+2}{z^2-2} = 3\)
43.
\(\dfrac{x}{x-2} = \dfrac{2}{x-2}+7\)
Answer.
No solution
44.
\(\dfrac{3x}{x+1} - \dfrac{2}{x^2+x} = \dfrac{4}{x}\)

Exercise Group.

For Problems 45–48, solve for the indicated variable.
45.
\(V=C\left(1-\dfrac{t}{n} \right) \text{,}\) for \(n\)
Answer.
\(n=\dfrac{Ct}{C-V} \)
46.
\(r = \dfrac{dc}{1-ec} \text{,}\) for \(c\)
47.
\(\dfrac{p}{q} = \dfrac{r}{q+r} \text{,}\) for \(q\)
Answer.
\(q=\dfrac{pr}{r-p} \)
48.
\(I = \dfrac{E}{R+\dfrac{r}{n}} \text{,}\) for \(R\)

Exercise Group.

For Problems 49–54, write the expression as a single fraction involving positive exponents only.
49.
\(x^{-3}+y^{-1}\)
Answer.
\(\dfrac{x^3+y}{x^3y}\)
50.
\(\dfrac{x^{-1}}{y}-\dfrac{x}{y^{-1}}\)
51.
\(\dfrac{x^{-1}-y}{x-y^{-1}}\)
Answer.
\(\dfrac{1-x^2y^2}{xy}\)
52.
\(\dfrac{x^{-1}+y^{-1}}{x^{-1}}\)
53.
\(\dfrac{x^{-1}-y^{-1}}{(x-y)^{-1}}\)
Answer.
\(\dfrac{-(x-y)^2}{xy}\)
54.
\(\dfrac{(xy)^{-1}}{x^{-1}-y^{-1}}\)
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