Mathematics
Rational Expressions
Rational expressions are mathematical expressions that can be written as a ratio of two polynomials. They can include variables and have the form P(x)/Q(x), where P(x) and Q(x) are polynomials and Q(x) is not equal to zero. Rational expressions can be simplified, added, subtracted, multiplied, and divided using techniques similar to those for rational numbers.
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eBook - PDFBeginning Algebra
Connecting Concepts through Applications
- Mark Clark, Cynthia Anfinson(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
The Basics of Rational Expressions and Equations LEARNING OBJECTIVES Evaluate Rational Expressions and equations. Find excluded values. Simplify Rational Expressions. 7.1 What’s That Mean? Rational The word ratio in mathematics refers to a quotient of two mathematical elements. A rational number is a ratio of integers, with a nonzero denominator. A rational expression is a ratio of polynomials. This is different from everyday language, in which saying that someone is rational means that the person is logical and sensible. Example 1 Cost per person for a ski trip A group of students is going to rent a cabin that costs $1000 for a weekend of skiing, and they plan to pay equal shares. The students can find the cost per person by using the equation c 5 1000 n where c is the cost in dollars per person for n students to go to the cabin for the weekend. a. Find the cost per person if five students stay in the cabin. b. When the cabin is full, students will pay the lowest cost per person. If the cabin can hold up to 20 people, what is the lowest cost per person? SOLUTION a. If five people stay in the cabin, substitute n 5 5 , and calculate the per-person cost. c 5 1000 5 5 20 0 If five students stay at the cabin, they will each have to pay $200. b. If the maximum number of people, 20, stay in the cabin, the lowest cost per person can be found by substituting n 5 20 and calculating the per person cost. c 5 1000 20 5 5 0 The lowest cost per person to rent the cabin is $50 when 20 students go to the cabin. DEFINITIONS Rational Expression An expression of the form P Q where P and Q are polynomials and Q ? 0 is called a rational expression. Rational Equation An equation that contains one or more Rational Expressions is called a rational equation. Rational Expression Rational Equation 2 x 1 5 x 2 4 4 x 2 7 3 x 1 2 5 5 Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
eBook - PDF- Mark D. Turner, Charles P. McKeague(Authors)
- 2016(Publication Date)
- XYZ Textbooks(Publisher)
Most people who have degrees in mathematics were students who could learn mathematics on their own. This doesn’t mean that you must learn it all on your own, or that you study alone, or that you don't ask questions. It means that you know your resources, both internal and external, and you can count on those resources when you need them. Attaining this goal gives you independence and puts you in control of your success in any math class you take. © clu/iStockPhoto This is the last chapter in which we will mention study skills. You know by now what works best for you and what you have to do to achieve your goals for this course. From now on, it is simply a matter of sticking with the things that work for you and avoiding the things that do not. It seems simple, but as with anything that takes effort, it is up to you to see that you maintain the skills that get you where you want to be in the course. 415 6.1 Learning Objectives In this section, we will learn how to: 1. Evaluate a rational expression. 2. Determine when a rational expression is undefined. 3. Reduce a rational expression to lowest terms. 4. Reduce a rational expression containing factors that are opposites. Evaluating and Reducing Rational Expressions Introduction We will begin this section with the definition of a rational expression. Recall from Chapter 1 that a rational number is any number that can be expressed as the ratio of two integers: Rational numbers = a __ b a and b are integers, b ≠ 0 We define a rational expression in a similar fashion. A rational expression is any expression that can be written in the form P __ Q where P and Q are polynomials and Q ≠ 0. rational expression DEFINITION Some examples of Rational Expressions are 2 x − 3 ______ x + 5 x 2 − 5 x − 6 _________ x 2 − 1 a − b _____ b − a Evaluating Rational Expressions To evaluate a rational expression means to find its value when any variables in the expression are replaced by specific numbers.
eBook - PDFComputer Algebra and Symbolic Computation
Elementary Algorithms
- Joel S. Cohen(Author)
- 2002(Publication Date)
- A K Peters/CRC Press(Publisher)
9. (a) Modify the Algebraic-expand algorithm so that it obtains the ex panded form when the input expressions include powers with fraction exponents. Assume that the transformations in (6.41) and (6.42) are included in automatic simplification. (b) Modify the Expand-power procedure in part (a) so that it also obtains the expansions using the decomposition of fraction powers in (6.49) and (6.50). Most computer algebra languages have an operator to compute [_ATJ. (In Maple and MuPAD use floor, and in Mathematica use Floor.) 6.5 General Rational Expressions In a mathematical sense, a rational expression is defined as a quotient of two polynomials. In this section we discuss the rational expression structure of an algebraic expression and describe an algorithm that transforms an expression to a particular rational form. Definition 6.46. (Mathematical Definition) Let S = { x i , ... , x m} be a set of generalized variables. An algebraic expression u is a general ratio nal expression (G R E) in S if it has the form u = p/q, where p and q are GPEs in S. Example 6.47 260 6. Structure of Polynomials and Rational Expressions For each example, we have given one possible choice for S. Notice that the definition is interpreted in a broad sense to include GPEs for which the denominator is understood to be 1. □ The Numerator and Denominator Operators. To determine if an expression is a GRE, we must define precisely the numerator and denominator of the expression. The Numerator and Denominator operators, which are used for this purpose, are defined by the following transformation rules. Definition 6.48. Let u be an algebraic expression. The Numerator and Denominator operators are defined in terms of the tree structure of an expression and are interpreted in the context of automatic simplification. Although the operators are adequate for our Example 6.49. Consider the expression
eBook - PDFIntermediate Algebra
Concepts and Graphs 2E
- Charles P. McKeague(Author)
- 2019(Publication Date)
- XYZ Textbooks(Publisher)
405 Chapter Outline 5.1 Basic Properties and Reducing to Lowest Terms 5.2 Multiplication and Division of Rational Expressions 5.3 Addition and Subtraction of Rational Expressions 5.4 Complex Fractions 5.5 Equations with Rational Expressions 5.6 Applications 5.7 Division of Polynomials 5 Rational Expressions and Rational Functions © iStockphoto/BrianAJackson I f you have ever put yourself on a weight loss diet, you know that you lose more weight at the beginning of the diet than you do later. If we let W(x) represent a person’s weight after x weeks on the diet, then the rational function W(x) = 80(2x + 15) __________ x + 6 is a mathematical model of the person’s weekly progress on a diet intended to take them from 200 pounds to about 160 pounds. Rational functions are good models for quantities that fall off rapidly to begin with, and then level off over time. The table shows some values for this function, along with the graph of this function. As you progress through this chapter, you will acquire an intuitive feel for these types of functions, and as a result, you will see why they are good models for situations such as dieting. Weeks Since Weight Starting Diet (Nearest Pound) 0 200 4 184 8 177 12 173 16 171 20 169 24 168 Weekly Weight Loss 4 0 8 12 16 20 24 160 165 170 175 180 185 190 195 200 x y Weeks Weight (lb) StudY SkIlls 406 The study skills for this chapter cover the way you approach new situations in mathematics. The first study skill is a point of view you hold about your natural instincts for what does and doesn’t work in mathematics. The second study skill gives you a way of testing your instincts. 1. Don’t Let Your Intuition Fool You As you become more experienced and more successful in mathematics you will be able to trust your mathematical intuition. For now, though, it can get in the way of your success. For example, if you ask some students to “subtract 3 from −5” they will answer −2 or 2.
eBook - PDF- Lynn Marecek, MaryAnne Anthony-Smith, Andrea Honeycutt Mathis(Authors)
- 2020(Publication Date)
- Openstax(Publisher)
If you miss a problem, go back to the section listed and review the material. Solve: 1 6 x + 1 2 = 1 3 . If you missed this problem, review Example 2.48. BE PREPARED : : 8.20 Solve: n 2 − 5n − 36 = 0. If you missed this problem, review Example 7.73. BE PREPARED : : 8.21 Solve for y in terms of x : 5x + 2y = 10 for y. If you missed this problem, review Example 2.65. After defining the terms expression and equation early in Foundations, we have used them throughout this book. We have simplified many kinds of expressions and solved many kinds of equations. We have simplified many Rational Expressions so far in this chapter. Now we will solve rational equations. The definition of a rational equation is similar to the definition of equation we used in Foundations. Rational Equation A rational equation is two Rational Expressions connected by an equal sign. You must make sure to know the difference between Rational Expressions and rational equations. The equation contains an equal sign. Rational Expression Rational Equation 1 8 x + 1 2 1 8 x + 1 2 = 1 4 y + 6 y 2 − 36 y + 6 y 2 − 36 = y + 1 1 n − 3 + 1 n + 4 1 n − 3 + 1 n + 4 = 15 n 2 + n − 12 Solve Rational Equations We have already solved linear equations that contained fractions. We found the LCD of all the fractions in the equation and then multiplied both sides of the equation by the LCD to “clear” the fractions. Here is an example we did when we worked with linear equations: Chapter 8 Rational Expressions and Equations 969 We multiplied both sides by the LCD. Then we distributed. We simplified—and then we had an equation with no fractions. Finally, we solved that equation. We will use the same strategy to solve rational equations. We will multiply both sides of the equation by the LCD. Then we will have an equation that does not contain Rational Expressions and thus is much easier for us to solve.
eBook - PDF- Jerome Kaufmann, Karen Schwitters, , , Jerome Kaufmann, Karen Schwitters(Authors)
- 2014(Publication Date)
- Cengage Learning EMEA(Publisher)
Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Chapter 4 • Rational Expressions 216 For Problems 1– 10, perform the indicated divisions of poly-nomials by monomials. (Objective 1) 1. 9 x 4 1 18 x 3 3 x 2. 12 x 3 2 24 x 2 6 x 2 3. 2 24 x 6 1 36 x 8 4 x 2 4. 2 35 x 5 2 42 x 3 2 7 x 2 5. 15 a 3 2 25 a 2 2 40 a 5 a 6. 2 16 a 4 1 32 a 3 2 56 a 2 2 8 a 7. 13 x 3 2 17 x 2 1 28 x 2 x 8. 14 xy 2 16 x 2 y 2 2 20 x 3 y 4 2 xy 9. 2 18 x 2 y 2 1 24 x 3 y 2 2 48 x 2 y 3 6 xy 10. 2 27 a 3 b 4 2 36 a 2 b 3 1 72 a 2 b 5 9 a 2 b 2 For Problems 11– 52, perform the indicated divisions. (Objective 1) 11. x 2 2 7 x 2 78 x 1 6 12. x 2 1 11 x 2 60 x 2 4 13. ( x 2 1 12 x 2 160) 4 ( x 2 8) 14. ( x 2 2 18 x 2 175) 4 ( x 1 7) 15. 2 x 2 2 x 2 4 x 2 1 16. 3 x 2 2 2 x 2 7 x 1 2 17. 15 x 2 1 22 x 2 5 3 x 1 5 18. 12 x 2 2 32 x 2 35 2 x 2 7 19. 3 x 3 1 7 x 2 2 13 x 2 21 x 1 3 20. 4 x 3 2 21 x 2 1 3 x 1 10 x 2 5 21. (2 x 3 1 9 x 2 2 17 x 1 6) 4 (2 x 2 1) 22. (3 x 3 2 5 x 2 2 23 x 2 7) 4 (3 x 1 1) 23. (4 x 3 2 x 2 2 2 x 1 6) 4 ( x 2 2) 24. (6 x 3 2 2 x 2 1 4 x 2 3) 4 ( x 1 1) 25. ( x 4 2 10 x 3 1 19 x 2 1 33 x 2 18) 4 ( x 2 6) 26. ( x 4 1 2 x 3 2 16 x 2 1 x 1 6) 4 ( x 2 3) 27. x 3 2 125 x 2 5 28. x 3 1 64 x 1 4 29. ( x 3 1 64) 4 ( x 1 1) 30. ( x 3 2 8) 4 ( x 2 4) 31. (2 x 3 2 x 2 6) 4 ( x 1 2) 32. (5 x 3 1 2 x 2 3) 4 ( x 2 2) 33. 4 a 2 2 8 ab 1 4 b 2 a 2 b 34. 3 x 2 2 2 xy 2 8 y 2 x 2 2 y 35. 4 x 3 2 5 x 2 1 2 x 2 6 x 2 2 3 x 36. 3 x 3 1 2 x 2 2 5 x 2 1 x 2 1 2 x 37. 8 y 3 2 y 2 2 y 1 5 y 2 1 y 38. 5 y 3 2 6 y 2 2 7 y 2 2 y 2 2 y 39.
eBook - PDF- Mark D. Turner, Charles P. McKeague(Authors)
- 2017(Publication Date)
- XYZ Textbooks(Publisher)
Solve number problems involving Rational Expressions. 2. Solve motion problems involving Rational Expressions. 3. Solve work problems involving Rational Expressions. In this section we will solve some word problems whose equations involve Rational Expressions. Like the other word problems we have encountered, the more you work with them, the easier they become. Number Problems One number is twice another. The sum of their reciprocals is 9 _ 2 . Find the two numbers. SOLUTION Let x represent the smaller number. The larger then must be 2 x . Their reciprocals are 1 _ x and 1 __ 2 x , respectively. An equation that describes the situation is: 1 __ x + 1 __ 2 x = 9 __ 2 We can multiply both sides by the LCD of 2 x and then solve the resulting equation: 2 x 1 __ x + 2 x 1 __ 2 x = 2 x 9 __ 2 2 + 1 = 9 x 3 = 9 x x = 3 __ 9 = 1 __ 3 The smaller number is 1 _ 3 . The other number is twice as large, or 2 _ 3 . If we add their reciprocals, we have: 3 __ 1 + 3 __ 2 = 6 __ 2 + 3 __ 2 = 9 __ 2 The solutions check with the original problem. Motion Problems Recall from Section 7.1 that if an object travels at a constant rate r for a specified time t , then the distance traveled is given by the rate equation Distance = Rate ⋅ Time d = rt If we know the distance traveled and the rate, then we can find the time by dividing the distance by the rate: Time = Distance _______ Rate t = d __ r EXAMPLE 1 VIDEO EXAMPLES SECTION 7.7 Applications 500 Chapter 7 Rational Expressions The next two examples use this version of the rate equation to solve problems involving motion. A boat travels 30 miles up a river in the same amount of time it takes to travel 50 miles down the same river. If the current is 5 miles per hour, what is the speed of the boat in still water? SOLUTION The easiest way to work a problem like this is with a table. The top row of the table is labeled with d for distance, r for rate, and t for time.
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