Jay Abramson; Sharon North

Learning Objectives

In this section, you will:

Simplify rational expressions.
Multiply rational expressions.
Divide rational expressions.
Add and subtract rational expressions.
Simplify complex rational expressions.

Corequisite Skills

Learning Objectives

Identify the skills leading to successful preparation for a college level mathematics exam.
Create a plan for success when taking mathematics exams.

Objective 1: Identify the skills leading to successful preparation for a college level mathematics exam.

Complete the following surveys by placing a checkmark in the a column for each strategy based on the frequency that you engaged in the strategy during your last academic term.

Exam Preparation Strategies	Always	Sometimes	Never
1. Rework each of the examples my instructor did in class.
2. Create note cards to help in memorizing important formulas and problem-solving strategies for the exam.
3. Create a study schedule for each math exam and begin to study for the exam at least one week prior to the date. Spaced practice over 5-7 days is much more effective than cramming material in 1-2 sessions.
4. Work the review exercises at the end of each chapter of the text.
5. Visit my instructor’s office hours when I need assistance in preparing for an exam.
6. Spend time on note interactions (see the section on Cornell notes) each day.
7. Create a practice test using the questions I identified in my class notes (see the section on Cornell notes) and take it the week before the exam.
8. Review each of the student learning objectives at the beginning of all sections covered on the exam and use this list as a checklist for exam preparation.
9. Ask your instructor how many questions will be on the exam and if they award partial credit for work shown.
10. Work through the practice test at the end of each chapter of the text.
11. Get a good night’s sleep the night before my exam.
12. Come to each exam prepared with a goal of earning an A.

Exam Day Behaviors and Strategies	Always	Sometimes	Never
13. Make sure to grab a healthy breakfast the day of the exam.
14. Arrive or log in early to class on exam days.
15. Keep my phone put away in my bag during exams to avoid distractions.
16. Try to relax and take a few deep breaths before beginning the exam.
17. Use a pencil so that I can make corrections neatly.
18. Read through all directions before beginning the exam.
19. Write formulas that are memorized in the margins, top or back of the test to reference when needed.
20. Scan through my entire test before beginning and start off working on a problem I am confident in solving.
21. Work each of the questions that I find easier first.
22. Keep track of time. Do a quick assessment of how much time should be spent on each question.
23. Try different approaches to solve when I get stuck on a problem.
24. Draw a diagram when solving an application problem.
25. Do some work on each question.
26. Work neatly and show all steps.
27. Make sure to attach units to final answers when units are given in the problem. (for example: cm, $, or feet/second)
28. Stay working for the entire class session or online exam session. If finished early I use the additional time to review my work and check answers.
29. Circle final answers or write each on the answer blank.

After the Exam Behaviors and Strategies	Always	Sometimes	Never
30. I work back through my exams after they are returned, writing corrections in another color or highlighting them for future reference.
31. Keep my old exams in a binder or notebook and use this assessment to review for my final exam.
32. Take responsibility for my exam performance and try to learn from the experience.
33. Reflect on the test taking experience and make a list for yourself on what to do differently next time.
34. Reflect on your feelings while taking the exam. Plan to replace any negative self-statements with positive ones on future exams.
35. Celebrate my success after doing well on an exam! Talk to a friend or family member about my progress.

Scoring
Total Number in Each Column
Scoring:	Always: 4 points each	Sometimes: 2 points each	Never: 0 points each
Total points:			0

Practice Makes Perfect

Practice: Identify the skills leading to successful preparation for a college level mathematics exam.

1.

Each of the behaviors or attitudes listed in the table above are associated with successful college mathematics exam preparation. This means that students who use these strategies or are open to these beliefs pass their college math courses. Compute your total score and share your score with your study group in class. Be supportive of your fellow students and offer encouragement!

Total score =__________

2.

Based on this survey, create a list of the top 5 test preparation and taking strategies that you currently utilize, and feel are most helpful to you.

3.

Based on this survey, create a list of the top 5 test preparation and taking strategies that interest you, and that you feel could be most helpful to you this term. Plan on implementing these strategies.

Objective 2: Create a plan for success when taking mathematics exams.

It’s important to take the opportunity to reflect on your past experiences in taking math exams as you begin a new term. We can learn a lot from these reflections and thus work toward developing a strategy for improvement.
In the table below list 5 challenges you have had in past math courses when taking an exam and list a possible solution that you could try this semester.

Challenge: Possible Solution:

1. ________

2. ________

3. ________

4. ________

5. ________
Develop your plan for success. Keep in mind the idea of mindsets and try to approach your test taking strategies with a growth mindset. Now is the time for growth as you begin a new term. Share your plan with your study group members.

A pastry shop has fixed costs of $$ 280$ per week and variable costs of $$ 9$ per box of pastries. The shop’s costs per week in terms of $x,$ the number of boxes made, is $280 + 9 x .$ We can divide the costs per week by the number of boxes made to determine the cost per box of pastries.

\frac{280 + 9 x}{x}

Notice that the result is a polynomial expression divided by a second polynomial expression. In this section, we will explore quotients of polynomial expressions.

Simplifying Rational Expressions

The quotient of two polynomial expressions is called a rational expression. We can apply the properties of fractions to rational expressions, such as simplifying the expressions by canceling common factors from the numerator and the denominator. To do this, we first need to factor both the numerator and denominator. Let’s start with the rational expression shown.

\frac{x^{2} + 8 x + 16}{x^{2} + 11 x + 28}

We can factor the numerator and denominator to rewrite the expression.

\frac{{(x + 4)}^{2}}{(x + 4) (x + 7)}

Then we can simplify that expression by canceling the common factor $(x + 4) .$

\frac{x + 4}{x + 7}

How To

Given a rational expression, simplify it.

Factor the numerator and denominator.
Cancel any common factors.

Example 1

Simplifying Rational Expressions

Simplify $\frac{x^{2} - 9}{x^{2} + 4 x + 3} .$

Solution

\begin{array}{l} \frac{(x + 3) (x - 3)}{(x + 3) (x + 1)} & Factor the numerator and the denominator . \\ \frac{x - 3}{x + 1} & Cancel common factor (x + 3) . \end{array}

Analysis

We can cancel the common factor because any expression divided by itself is equal to 1.

Q&A

Can the $x^{2}$ term be cancelled in Example 1?

No. A factor is an expression that is multiplied by another expression. The $x^{2}$ term is not a factor of the numerator or the denominator.

Try It #1

Simplify $\frac{x - 6}{x^{2} - 36} .$

Multiplying Rational Expressions

Multiplication of rational expressions works the same way as multiplication of any other fractions. We multiply the numerators to find the numerator of the product, and then multiply the denominators to find the denominator of the product. Before multiplying, it is helpful to factor the numerators and denominators just as we did when simplifying rational expressions. We are often able to simplify the product of rational expressions.

How To

Given two rational expressions, multiply them.

Factor the numerator and denominator.
Multiply the numerators.
Multiply the denominators.
Simplify.

Example 2

Multiplying Rational Expressions

Multiply the rational expressions and show the product in simplest form:

\frac{x^{2} + 4 x - 5}{3 x + 18} \cdot \frac{2 x - 1}{x + 5}

Solution

\begin{array}{l} \frac{(x + 5) (x - 1)}{3 (x + 6)} \cdot \frac{(2 x - 1)}{(x + 5)} & Factor the numerator and denominator . \\ \frac{(x + 5) (x - 1) (2 x - 1)}{3 (x + 6) (x + 5)} & Multiply numerators and denominators . \\ \frac{(x + 5) (x - 1) (2 x - 1)}{3 (x + 6) (x + 5)} & Cancel common factors to simplify . \\ \frac{(x - 1) (2 x - 1)}{3 (x + 6)} \end{array}

Try It #2

Multiply the rational expressions and show the product in simplest form:

\frac{x^{2} + 11 x + 30}{x^{2} + 5 x + 6} \cdot \frac{x^{2} + 7 x + 12}{x^{2} + 8 x + 16}

Dividing Rational Expressions

Division of rational expressions works the same way as division of other fractions. To divide a rational expression by another rational expression, multiply the first expression by the reciprocal of the second. Using this approach, we would rewrite $\frac{1}{x} \div \frac{x^{2}}{3}$ as the product $\frac{1}{x} \cdot \frac{3}{x^{2}} .$ Once the division expression has been rewritten as a multiplication expression, we can multiply as we did before.

\frac{1}{x} \cdot \frac{3}{x^{2}} = \frac{3}{x^{3}}

How To

Given two rational expressions, divide them.

Rewrite as the first rational expression multiplied by the reciprocal of the second.
Factor the numerators and denominators.
Multiply the numerators.
Multiply the denominators.
Simplify.

Example 3

Dividing Rational Expressions

Divide the rational expressions and express the quotient in simplest form:

\frac{2 x^{2} + x - 6}{x^{2} - 1} \div \frac{x^{2} - 4}{x^{2} + 2 x + 1}

Solution

\begin{matrix} \frac{2 x^{2} + x - 6}{x^{2} - 1} \cdot \frac{x^{2} + 2 x + 1}{x^{2} - 4} & Rewrite as multiplication. \\ \frac{(2 x - 3) (x + 2)}{(x + 1) (x - 1)} \cdot \frac{{(x + 1)}^{2}}{(x + 2) (x - 2)} & Factor. \\ \frac{(2 x - 3) (x + 2) {(x + 1)}^{2}}{(x + 1) (x - 1) (x + 2) (x - 2)} & Multiply. \\ \frac{(2 x - 3) (x + 1)}{(x - 1) (x - 2)} & Cancel common factors to simplify. \end{matrix}

Try It #3

Divide the rational expressions and express the quotient in simplest form:

\frac{9 x^{2} - 16}{3 x^{2} + 17 x - 28} \div \frac{3 x^{2} - 2 x - 8}{x^{2} + 5 x - 14}

Adding and Subtracting Rational Expressions

Adding and subtracting rational expressions works just like adding and subtracting numerical fractions. To add fractions, we need to find a common denominator. Let’s look at an example of fraction addition.

\begin{matrix} \frac{5}{24} + \frac{1}{40} & = & \frac{25}{120} + \frac{3}{120} \\ = & \frac{28}{120} \\ = & \frac{7}{30} \end{matrix}

We have to rewrite the fractions so they share a common denominator before we are able to add. We must do the same thing when adding or subtracting rational expressions.

The easiest common denominator to use will be the least common denominator, or LCD. The LCD is the smallest multiple that the denominators have in common. To find the LCD of two rational expressions, we factor the expressions and multiply all of the distinct factors. For instance, if the factored denominators were $(x + 3) (x + 4)$ and $(x + 4) (x + 5),$ then the LCD would be $(x + 3) (x + 4) (x + 5) .$

Once we find the LCD, we need to multiply each expression by the form of 1 that will change the denominator to the LCD. We would need to multiply the expression with a denominator of $(x + 3) (x + 4)$ by $\frac{x + 5}{x + 5}$ and the expression with a denominator of $(x + 4) (x + 5)$ by $\frac{x + 3}{x + 3} .$

How To

Given two rational expressions, add or subtract them.

Factor the numerator and denominator.
Find the LCD of the expressions.
Multiply the expressions by a form of 1 that changes the denominators to the LCD.
Add or subtract the numerators.
Simplify.

Example 4

Adding Rational Expressions

Add the rational expressions:

\frac{5}{x} + \frac{6}{y}

Solution

First, we have to find the LCD. In this case, the LCD will be $x y .$ We then multiply each expression by the appropriate form of 1 to obtain $x y$ as the denominator for each fraction.

\begin{array}{l} \frac{5}{x} \cdot \frac{y}{y} + \frac{6}{y} \cdot \frac{x}{x} \\ \frac{5 y}{x y} + \frac{6 x}{x y} \end{array}

Now that the expressions have the same denominator, we simply add the numerators to find the sum.

\frac{6 x + 5 y}{x y}

Analysis

Multiplying by $\frac{y}{y}$ or $\frac{x}{x}$ does not change the value of the original expression because any number divided by itself is 1, and multiplying an expression by 1 gives the original expression.

Example 5

Subtracting Rational Expressions

Subtract the rational expressions:

\frac{6}{x^{2} + 4 x + 4} - \frac{2}{x^{2} −4}

Solution

\begin{matrix} \frac{6}{{(x + 2)}^{2}} - \frac{2}{(x + 2) (x - 2)} & Factor . \\ \frac{6}{{(x + 2)}^{2}} \cdot \frac{x - 2}{x - 2} - \frac{2}{(x + 2) (x - 2)} \cdot \frac{x + 2}{x + 2} & Multiply each fraction to get LCD as denominator . \\ \frac{6 (x - 2)}{{(x + 2)}^{2} (x - 2)} - \frac{2 (x + 2)}{{(x + 2)}^{2} (x - 2)} & Multiply . \\ \frac{6 x - 12 - (2 x + 4)}{{(x + 2)}^{2} (x - 2)} & Apply distributive property . \\ \frac{4 x - 16}{{(x + 2)}^{2} (x - 2)} & Subtract . \\ \frac{4 (x - 4)}{{(x + 2)}^{2} (x - 2)} & Simplify . \end{matrix}

Q&A

Do we have to use the LCD to add or subtract rational expressions?

No. Any common denominator will work, but it is easiest to use the LCD.

Try It #4

Subtract the rational expressions: $\frac{3}{x + 5} - \frac{1}{x −3} .$

Simplifying Complex Rational Expressions

A complex rational expression is a rational expression that contains additional rational expressions in the numerator, the denominator, or both. We can simplify complex rational expressions by rewriting the numerator and denominator as single rational expressions and dividing. The complex rational expression $\frac{a}{\frac{1}{b} + c}$ can be simplified by rewriting the numerator as the fraction $\frac{a}{1}$ and combining the expressions in the denominator as $\frac{1 + b c}{b} .$ We can then rewrite the expression as a multiplication problem using the reciprocal of the denominator. We get $\frac{a}{1} \cdot \frac{b}{1 + b c},$ which is equal to $\frac{a b}{1 + b c} .$

How To

Given a complex rational expression, simplify it.

Combine the expressions in the numerator into a single rational expression by adding or subtracting.
Combine the expressions in the denominator into a single rational expression by adding or subtracting.
Rewrite as the numerator divided by the denominator.
Rewrite as multiplication.
Multiply.
Simplify.

Example 6

Simplifying Complex Rational Expressions

Simplify: $\frac{y + \frac{1}{x}}{\frac{x}{y}}$ .

Solution

Begin by combining the expressions in the numerator into one expression.

\begin{matrix} y \cdot \frac{x}{x} + \frac{1}{x} & Multiply by \frac{x}{x} to get LCD as denominator . \\ \frac{x y}{x} + \frac{1}{x} \\ \frac{x y + 1}{x} & Add numerators . \end{matrix}

Now the numerator is a single rational expression and the denominator is a single rational expression.

\frac{\frac{x y + 1}{x}}{\frac{x}{y}}

We can rewrite this as division, and then multiplication.

\begin{matrix} \frac{x y + 1}{x} \div \frac{x}{y} \\ \frac{x y + 1}{x} \cdot \frac{y}{x} & Rewrite as multiplication . \\ \frac{y (x y + 1)}{x^{2}} & Multiply . \end{matrix}

Try It #5

Simplify: $\frac{\frac{x}{y} - \frac{y}{x}}{y}$

Q&A

Can a complex rational expression always be simplified?

Yes. We can always rewrite a complex rational expression as a simplified rational expression.

Media

Access these online resources for additional instruction and practice with rational expressions.

1.6 Section Exercises

Verbal

1.

How can you use factoring to simplify rational expressions?

2.

How do you use the LCD to combine two rational expressions?

3.

Tell whether the following statement is true or false and explain why: You only need to find the LCD when adding or subtracting rational expressions.

Algebraic

For the following exercises, simplify the rational expressions.

4.

$\frac{x^{2} - 16}{x^{2} - 5 x + 4}$

5.

$\frac{y^{2} + 10 y + 25}{y^{2} + 11 y + 30}$

6.

$\frac{6 a^{2} - 24 a + 24}{6 a^{2} - 24}$

7.

$\frac{9 b^{2} + 18 b + 9}{3 b + 3}$

8.

$\frac{m - 12}{m^{2} - 144}$

9.

$\frac{2 x^{2} + 7 x - 4}{4 x^{2} + 2 x - 2}$

10.

$\frac{6 x^{2} + 5 x - 4}{3 x^{2} + 19 x + 20}$

11.

$\frac{a^{2} + 9 a + 18}{a^{2} + 3 a - 18}$

12.

$\frac{3 c^{2} + 25 c - 18}{3 c^{2} - 23 c + 14}$

13.

$\frac{12 n^{2} - 29 n - 8}{28 n^{2} - 5 n - 3}$

For the following exercises, multiply the rational expressions and express the product in simplest form.

14.

$\frac{x^{2} - x - 6}{2 x^{2} + x - 6} \cdot \frac{2 x^{2} + 7 x - 15}{x^{2} - 9}$

15.

$\frac{c^{2} + 2 c - 24}{c^{2} + 12 c + 36} \cdot \frac{c^{2} - 10 c + 24}{c^{2} - 8 c + 16}$

16.

$\frac{2 d^{2} + 9 d - 35}{d^{2} + 10 d + 21} \cdot \frac{3 d^{2} + 2 d - 21}{3 d^{2} + 14 d - 49}$

17.

$\frac{10 h^{2} - 9 h - 9}{2 h^{2} - 19 h + 24} \cdot \frac{h^{2} - 16 h + 64}{5 h^{2} - 37 h - 24}$

18.

$\frac{6 b^{2} + 13 b + 6}{4 b^{2} - 9} \cdot \frac{6 b^{2} + 31 b - 30}{18 b^{2} - 3 b - 10}$

19.

$\frac{2 d^{2} + 15 d + 25}{4 d^{2} - 25} \cdot \frac{2 d^{2} - 15 d + 25}{25 d^{2} - 1}$

20.

$\frac{6 x^{2} - 5 x - 50}{15 x^{2} - 44 x - 20} \cdot \frac{20 x^{2} - 7 x - 6}{2 x^{2} + 9 x + 10}$

21.

$\frac{t^{2} - 1}{t^{2} + 4 t + 3} \cdot \frac{t^{2} + 2 t - 15}{t^{2} - 4 t + 3}$

22.

$\frac{2 n^{2} - n - 15}{6 n^{2} + 13 n - 5} \cdot \frac{12 n^{2} - 13 n + 3}{4 n^{2} - 15 n + 9}$

23.

$\frac{36 x^{2} - 25}{6 x^{2} + 65 x + 50} \cdot \frac{3 x^{2} + 32 x + 20}{18 x^{2} + 27 x + 10}$

For the following exercises, divide the rational expressions.

24.

$\frac{3 y^{2} - 7 y - 6}{2 y^{2} - 3 y - 9} \div \frac{y^{2} + y - 2}{2 y^{2} + y - 3}$

25.

$\frac{6 p^{2} + p - 12}{8 p^{2} + 18 p + 9} \div \frac{6 p^{2} - 11 p + 4}{2 p^{2} + 11 p - 6}$

26.

$\frac{q^{2} - 9}{q^{2} + 6 q + 9} \div \frac{q^{2} - 2 q - 3}{q^{2} + 2 q - 3}$

27.

$\frac{18 d^{2} + 77 d - 18}{27 d^{2} - 15 d + 2} \div \frac{3 d^{2} + 29 d - 44}{9 d^{2} - 15 d + 4}$

28.

$\frac{16 x^{2} + 18 x - 55}{32 x^{2} - 36 x - 11} \div \frac{2 x^{2} + 17 x + 30}{4 x^{2} + 25 x + 6}$

29.

$\frac{144 b^{2} - 25}{72 b^{2} - 6 b - 10} \div \frac{18 b^{2} - 21 b + 5}{36 b^{2} - 18 b - 10}$

30.

$\frac{16 a^{2} - 24 a + 9}{4 a^{2} + 17 a - 15} \div \frac{16 a^{2} - 9}{4 a^{2} + 11 a + 6}$

31.

$\frac{22 y^{2} + 59 y + 10}{12 y^{2} + 28 y - 5} \div \frac{11 y^{2} + 46 y + 8}{24 y^{2} - 10 y + 1}$

32.

$\frac{9 x^{2} + 3 x - 20}{3 x^{2} - 7 x + 4} \div \frac{6 x^{2} + 4 x - 10}{x^{2} - 2 x + 1}$

For the following exercises, add and subtract the rational expressions, and then simplify.

33.

$\frac{4}{x} + \frac{10}{y}$

34.

$\frac{12}{2 q} - \frac{6}{3 p}$

35.

$\frac{4}{a + 1} + \frac{5}{a - 3}$

36.

$\frac{c + 2}{3} - \frac{c - 4}{4}$

37.

$\frac{y + 3}{y - 2} + \frac{y - 3}{y + 1}$

38.

$\frac{x - 1}{x + 1} - \frac{2 x + 3}{2 x + 1}$

39.

$\frac{3 z}{z + 1} + \frac{2 z + 5}{z - 2}$

40.

$\frac{4 p}{p + 1} - \frac{p + 1}{4 p}$

41.

$\frac{x}{x + 1} + \frac{y}{y + 1}$

For the following exercises, simplify the rational expression.

42.

$\frac{\frac{6}{y} - \frac{4}{x}}{y}$

43.

$\frac{\frac{2}{a} + \frac{7}{b}}{b}$

44.

$\frac{\frac{x}{4} - \frac{p}{8}}{p}$

45.

$\frac{\frac{3}{a} + \frac{b}{6}}{\frac{2 b}{3 a}}$

46.

$\frac{\frac{3}{x + 1} + \frac{2}{x - 1}}{\frac{x - 1}{x + 1}}$

47.

$\frac{\frac{a}{b} - \frac{b}{a}}{\frac{a + b}{a b}}$

48.

$\frac{\frac{2 x}{3} + \frac{4 x}{7}}{\frac{x}{2}}$

49.

$\frac{\frac{2 c}{c + 2} + \frac{c - 1}{c + 1}}{\frac{2 c + 1}{c + 1}}$

50.

$\frac{\frac{x}{y} - \frac{y}{x}}{\frac{x}{y} + \frac{y}{x}}$

Real-World Applications

51.

Brenda is placing tile on her bathroom floor. The area of the floor is $15 x^{2} - 8 x - 7$ ft². The area of one tile is $x^{2} - 2 x + 1 {ft}^{2} .$ To find the number of tiles needed, simplify the rational expression: $\frac{15 x^{2} - 8 x - 7}{x^{2} - 2 x + 1} .$

A rectangle that’s labeled: Area = fifteen times x squared minus eight times x minus seven.

52.

The area of Lijuan's yard is $25 x^{2} - 625$ ft². A patch of sod has an area of $x^{2} - 10 x + 25$ ft². Divide the two areas and simplify to find how many pieces of sod Lijuan needs to cover her yard.

53.

Elroi wants to mulch his garden. His garden is $x^{2} + 18 x + 81$ ft². One bag of mulch covers $x^{2} - 81$ ft². Divide the expressions and simplify to find how many bags of mulch Elroi needs to mulch his garden.

Extensions

For the following exercises, perform the given operations and simplify.

54.

$\frac{x^{2} + x - 6}{x^{2} - 2 x - 3} \cdot \frac{2 x^{2} - 3 x - 9}{x^{2} - x - 2} \div \frac{10 x^{2} + 27 x + 18}{x^{2} + 2 x + 1}$

55.

$\frac{\frac{3 y^{2} - 10 y + 3}{3 y^{2} + 5 y - 2} \cdot \frac{2 y^{2} - 3 y - 20}{2 y^{2} - y - 15}}{y - 4}$

56.

$\frac{\frac{4 a + 1}{2 a - 3} + \frac{2 a - 3}{2 a + 3}}{\frac{4 a^{2} + 9}{a}}$

57.

$\frac{x^{2} + 7 x + 12}{x^{2} + x - 6} \div \frac{3 x^{2} + 19 x + 28}{8 x^{2} - 4 x - 24} \div \frac{2 x^{2} + x - 3}{3 x^{2} + 4 x - 7}$

Challenge:	Possible Solution:
1. ________
2. ________
3. ________
4. ________
5. ________

1.6 Rational Expressions

Corequisite Skills

Learning Objectives

Objective 1: Identify the skills leading to successful preparation for a college level mathematics exam.

Practice Makes Perfect

Objective 2: Create a plan for success when taking mathematics exams.

Simplifying Rational Expressions

Simplifying Rational Expressions

Solution

Analysis

Multiplying Rational Expressions

Multiplying Rational Expressions

Solution

Dividing Rational Expressions

Dividing Rational Expressions

Solution

Adding and Subtracting Rational Expressions

Adding Rational Expressions

Solution

Analysis

Subtracting Rational Expressions

Solution

Simplifying Complex Rational Expressions

Simplifying Complex Rational Expressions

Solution

1.6 Section Exercises

Verbal

Algebraic

Real-World Applications

Extensions