CHAPTER 6 Rational Epressions 6. Rational Functions and Multiplying and Dividing Rational Epressions 6. Adding and Subtracting Rational Epressions 6.3 Simplifying Comple Fractions 6. Dividing Polynomials: Long Division and Synthetic Division 6. Solving Equations Containing Rational Epressions Integrated Review Epressions and Equations Containing Rational Epressions 6.6 Rational Equations and Problem Solving During the past 0 years, music has changed from a physical album, tape, or CD to a digital form. This has enabled consumers to download music and move it among different players. In the year 0, it is predicted that consumer spending on CDs will fall below consumer spending on digital music. In the Chapter Review, Eercises 07 and 08, we will review the future of digital sales. Billions of Dollars.0.0 3.0.0.0 0 Source: RIAA 0.. U.S. Digital Music Sales.9.7 3.0 3. 3. 00 00 006 007 008 009 00 03* Year 3. *Forecasted 6.7 Variation and Problem Solving Polynomials are to algebra what integers are to arithmetic. We have added, subtracted, multiplied, and raised polynomials to powers, each operation yielding another polynomial, just as these operations on integers yield another integer. But when we divide one integer by another, the result may or may not be another integer. Likewise, when we divide one polynomial by another, we may or may not get a polynomial in return. The quotient, + is not a polynomial; it is a rational epression that can be written as +. In this chapter, we study these new algebraic forms known as rational epressions and the rational functions they generate. 33
336 CHAPTER 6 Rational Epressions 6. Rational Functions and Multiplying and Dividing Rational Epressions S Find the Domain of a Rational Epression. Simplify Rational Epressions. 3 Multiply Rational Epressions. Divide Rational Epressions. Use Rational Functions in Applications. Recall that a rational number, or fraction, is a number that can be written as the quotient p of two integers p and q as long as q is not 0. A rational epression is an epression that q can be written as the quotient P of two polynomials P and Q as long as Q is not 0. Q Eamples of Rational Epressions 3 + 7-3 - 7 - + 7 + 6 Rational epressions are sometimes used to describe functions. For eample, we call the function f + - 3 a rational function since + is a rational epression. - 3 Finding the Domain of a Rational Epression As with fractions, a rational epression is undefined if the denominator is 0. If a variable in a rational epression is replaced with a number that makes the denominator 0, we say that the rational epression is undefined for this value of the variable. For eample, the rational epression + is undefined when is 3 because replacing - 3 with 3 results in a denominator of 0. For this reason, we must eclude 3 from the domain of the function f + - 3. The domain of f is then { is a real number and Z 3} The set of all such that is a real number and is not equal to 3. In this section, we will use this set builder notation to write domains. Unless told otherwise, we assume that the domain of a function described by an equation is the set of all real numbers for which the equation is defined. EXAMPLE f 83 + 7 + 0 Find the domain of each rational function. c. We find the domain by setting the denominator equal to 0. - - 0 - + 3 0 g - 3 - c. f 7 - - - The domain of each function will contain all real numbers ecept those values that make the denominator 0. No matter what the value of, the denominator of f 83 + 7 + 0 is never 0, so the domain of f is 0 is a real number6. To find the values of that make the denominator of g() equal to 0, we solve the equation ;denominator 0<: - 0 or The domain must eclude since the rational epression is undefined when is. The domain of g is 0 is a real number and 6. Set the denominator equal to 0 and solve.
Section 6. Rational Functions and Multiplying and Dividing Rational Epressions 337 0-0 or + 3 0 or -3 If is replaced with or with -3, the rational epression is undefined. The domain of f is is a real number and, -36. Find the domain of each rational function. f - 3 + -6 g 6 + + 3 c. h 8-3 - + 6 CONCEPT CHECK For which of these values (if any) is the rational epression - 3 + undefined? 3 c. - d. 0 e. None of these Simplifying Rational Epressions Recall that a fraction is in lowest terms or simplest form if the numerator and denominator have no common factors other than (or -). For eample, 3 is in lowest terms 3 since 3 and 3 have no common factors other than (or -). To simplify a rational epression, or to write it in lowest terms, we use a method similar to simplifying a fraction. Recall that to simplify a fraction, we essentially remove factors of. Our ability to do this comes from these facts: If c 0, then c c. For eample, 7-8.6 and 7-8.6. n n. For eample, - -, 6.8 a 6.8, and a, b 0. b b In other words, we have the following: a b c c a b c c a b Since a b a b Let s practice simplifying a fraction by simplifying 6. 3 3 3 3 6 3 3 3 3 Answer to Concept Check: e Let s use the same technique and simplify the rational epression + - + + - + + + - + + - + - + -.
338 CHAPTER 6 Rational Epressions + This means that the rational epression has the same value as the rational - epression + for all values of ecept and -. (Remember that when is, the - denominators of both rational epressions are 0 and that when is -, the original rational epression has a denominator of 0.) As we simplify rational epressions, we will assume that the simplified rational epression is equivalent to the original rational epression for all real numbers ecept those for which either denominator is 0. Just as for numerical fractions, we can use a shortcut notation. Remember that as long as eact factors in both the numerator and denominator are divided out, we are removing a factor of. We can use the following notation: + - + + - + + - A factor of is identified by the shading. Remove the factor of. This removing a factor of is stated in the principle below: Fundamental Principle of Rational Epressions For any rational epression P and any polynomial R, where R 0, Q or, simply, PR QR P Q R R P Q P Q PR QR P Q In general, the following steps may be used to simplify rational epressions or to write a rational epression in lowest terms. Simplifying or Writing a Rational Epression in Lowest Terms Step. Completely factor the numerator and denominator of the rational epression. Step. Divide out factors common to the numerator and denominator. (This is the same as removing a factor of. ) For now, we assume that variables in a rational epression do not represent values that make the denominator 0. EXAMPLE Simplify each rational epression. 0 3-9 + 3 + 8 + - 7 0 3 - - - -
Section 6. Rational Functions and Multiplying and Dividing Rational Epressions 339 9 + 3 + 8 + - 7 9 + + 8-7 + 9 + 8-7 9 + 8-7 Factor the numerator and denominator. Since + + Simplest form Simplify each rational epressions. z 0z - z + 3 + 6 6 + 7-0 Helpful Hint When the numerator and the denominator of a rational epression are opposites of each other, the epression simplifies to -. EXAMPLE 3 + + Simplify each rational epression. - - + + + By the commutative property of addition, + +. + - - The terms in the numerator of - differ by sign from the terms of the denominator, - so the polynomials are opposites of each other and the epression simplifies to -. To see this, we factor out - from the numerator or the denominator. If - is factored from the numerator, then - - - - + - - - - If - is factored from the denominator, the result is the same. - - - - - + - - - - - - - 3 Simplify each rational epression. + 3 3 + 3 - - 3 EXAMPLE Simplify 8 - - - 3. 8 - - - 3 9 - + - 3 3 + 3 - + - 3 3 + - - 3 + - 3 3 + - + Factor. Factor completely. Notice the opposites 3 - and - 3. Write 3 - as - - 3 and simplify. 0 - Simplify + - 6.
30 CHAPTER 6 Rational Epressions Helpful Hint Recall that for a fraction, For eample - + + a -b -a b - a b + - + - + + CONCEPT CHECK Which of the following epressions are equivalent to - - 8-8 - c. - 8 8 -? d. - -8 + EXAMPLE Simplify each rational epression. 3 + 8 y + + y 3 - y + y - 3 + 8 + + - + + - + y + y 3 - y + y - y + y 3 - y + y - y + y y - + y - y + y - y + y - Simplify each rational epression. 3 + 6 z + 0 + z 3-3z + z - 6 Factor the sum of the two cubes. Divide out common factors. Factor the numerator. Factor the denominator by grouping. Divide out common factors. CONCEPT CHECK n Does n + simplify to? Why or why not? Answers to Concept Checks: a and d no; answers may vary. 3 Multiplying Rational Epressions Arithmetic operations on rational epressions are performed in the same way as they are on rational numbers.
Section 6. Rational Functions and Multiplying and Dividing Rational Epressions 3 Multiplying Rational Epressions The rule for multiplying rational epressions is P R Q S PR as long as Q 0 and S 0. QS To multiply rational epressions, you may use these steps: Step. Step. Step 3. Completely factor each numerator and denominator. Use the rule above and multiply the numerators and the denominators. Simplify the product by dividing the numerator and denominator by their common factors. When we multiply rational epressions, notice that we factor each numerator and denominator first. This helps when we apply the fundamental principle to write the product in simplest form. EXAMPLE 6 + 3n n + 3n n Multiply. n - 3n - n - 3 - -3 + 3 + + n - 3n - n - + 3n n - n 3n + n - + 3n n - n 3n + n - n - nn - 3 - -3 + 3 + + - + + 3-3 - + + - + + 3-3 - + + - - Factor. Multiply. Divide out common factors. Factor. Multiply. Simplest form 6 Multiply. + n 3n 6n + 3 n - 3n - 3-8 6-6 + + + Dividing Rational Epressions Recall that two numbers are reciprocals of each other if their product is. Similarly, if P Q is a rational epression, then Q is its reciprocal, since P P Q Q P P Q Q P
3 CHAPTER 6 Rational Epressions The following are eamples of epressions and their reciprocals. Epression 3 + - 3 3 0 Reciprocal 3-3 + 3 no reciprocal Dividing Rational Epressions The rule for dividing rational epressions is P Q, R S P S Q R PS as long as Q 0, S 0, and R 0. QR To divide by a rational epression, use the rule above and multiply by its reciprocal. Then simplify if possible. Notice that division of rational epressions is the same as for rational numbers. EXAMPLE 7 Divide. 8m 3m -, 0 - m 8y + 9y - y - 0y +, 3y + 7y + 0 8y + 8y - 8m 3m -, 0 - m 8m - m 3m - 0-8m - m 3m + m - 0 8m - m - 3m + m - 8 m m + Multiply by the reciprocal of the divisor. Factor and multiply. Write - m as -m -. Simplify. 8y + 9y - y - 0y +, 3y + 7y + 0 8y + 8y - 8y + 9y - 8y + 8y - y - 0y + 3y + 7y + 0 Multiply by the reciprocal. 6y - 3y + 6y - y - y - y + 3y + y + Factor. y + y + Simplest form 7 Divide. 6y 3 3y - 7, 3 - y 0 + 3 - - + 0, + 9 + 0 7-68 - 0
Section 6. Rational Functions and Multiplying and Dividing Rational Epressions 33 Helpful Hint When dividing rational epressions, do not divide out common factors until the division problem is rewritten as a multiplication problem. EXAMPLE 8 Perform each indicated operation. - 3 +, - 3-0 + - 3 + +, - 3-0 - 3 + + - 3-0 + - + + 3 + 3 + To divide, multiply by the reciprocal - + 8 Perform each indicated operation. - 6-0 - 3, + - Using Rational Functions in Applications Rational functions occur often in real-life situations. EXAMPLE 9 Cost for Pressing Compact Discs.6 + 0,000 For the ICL Production Company, the rational function C describes the company s cost per disc for pressing compact discs. Find the cost per disc for pressing: 00 compact discs 000 compact discs.600 + 0,000 C00 0,60 0.6 00 00 The cost per disc for pressing 00 compact discs is $0.60..6000 + 0,000 C000,600 000 000.6 The cost per disc for pressing 000 compact discs is $.60. Notice that as more compact discs are produced, the cost per disc decreases. 9 A company s cost per tee shirt for silk screening tee shirts is given by the 3. + 00 rational function C. Find the cost per tee shirt for printing: 00 tee shirts 000 tee shirts
3 CHAPTER 6 Rational Epressions Graphing Calculator Eplorations 0 0 0 0 0 0 0 0 y 3 8 7 6 3 3 3 7 - Recall that since the rational epression is not defined - + when or when -, we say that the domain of the rational function 7 - f is all real numbers ecept and -. This domain can be - + written as 0 is a real number and, -6. This means that the graph of f() should not cross the vertical lines and -. The graph of f() in connected mode is to the left. In connected mode, the graphing calculator tries to connect all dots of the graph so that the result is a smooth curve. This is what has happened in the graph. Notice that the graph appears to contain vertical lines at and at -. We know that this cannot happen because the function is not defined at and at -. We also know that this cannot happen because the graph of this function would not pass the vertical line test. The graph of f() in dot mode is to the left. In dot mode, the graphing calculator will not connect dots with a smooth curve. Notice that the vertical lines have disappeared, and we have a better picture of the graph. The graph, however, actually appears more like the hand-drawn graph below. By using a Table feature, a Calculate Value feature, or by tracing, we can see that the function is not defined at and at -. Find the domain of each rational function. Then graph each rational function and use the graph to confirm the domain.. f + -. g 3. h. f - 9 + 7-3 + - 9 - Vocabulary, Readiness & Video Check Use the choices below to fill in each blank. Some choices may not be used. true rational simplified -a -a -b b - false domain 0 a -b. A epression is an epression that can be written as the quotient P of two polynomials P and Q Q as long as Q 0.. A rational epression is undefined if the denominator is. 3. The of the rational function f is { is a real number and 0}.. A rational epression is if the numerator and denominator have no common factors other than or -.. The epression + simplifies to. + 6. The epression y - z simplifies to. z - y
Section 6. Rational Functions and Multiplying and Dividing Rational Epressions 3 7. For a rational epression, - a b. 8. True or false: a - 6 a + -a - 6 -a + -a + 6 -a -. Martin-Gay Interactive Videos See Video 6. Watch the section lecture video and answer the following questions. 3 9. From Eamples and, why can t the denominators of rational epressions be zero? How does this relate to the domain of a rational function? 0. What conclusion does Eample lead us to?. Does a person need to be comfortable with factoring polynomials in order to be successful with multiplying rational epressions? Eplain, referencing Eample 6 in your answer.. Based on Eample 7, complete the following statements. Division of rational epressions is similar to division of. Therefore, to divide rational epressions, it s the first rational epression times the of the second rational epression. 3. In Eample 8d, we find the domain of the function. Do you think the domain should be restricted in any way given the contet of this application? Eplain. 6. Eercise Set Find the domain of each rational function. See Eample.. f - 7. g - 3 3. st t + t. vt - t + t 3t. f 3 7 - - 6. f - + 7. f 3 - - 8. g + 3 + 9. R 3 + - - 3 0. h - + 0. C + 3 -. R - 7 Simplify each rational epression. See Eamples through. 3. 8-6 3-6. 8 3. - 9-6. 3 + + 7. 9y - 8 6y - 8 8. 7y - y - 6 9. + 6-0 - 8 + 6 0. + 0 -. - 9 -. 9 - - 3. - 9 - y. 7 - y -. - 7-3 - + 0 6. - + - 7 + 0 7. 3 - + 8. - 0 3 + 9. 3 - - - - 3 6 3 + 30. + 3 + 3-3 + - 3 9 - + 8 3-7 3. 7 3 3. + + 6 + 9 Multiply and simplify. See Eample 6. 33. 3. - 0-7 6 - -
36 CHAPTER 6 Rational Epressions 3. 36. 37. 38. 39. 8a - a a + a + a - b a + ab b - a 0b - a 9 + 9 + + 8 3-3 - + 36 0 + 30 3-3 3-6 6 + 6-36 a + 8a + 3 a - 9 9 + 8 3 + 6 + 0. - 3 + 9-9 - 0-0 3 + 7. a 3 + a b + a + b 6a a 3 + a a - b. a - 8a 8b + ab - b + 3a - 6 3a + 6 3. - 6-6 + 6 + 6-8 3 + 30 + 8. + - 3 + 0 + 6 + 6 + 6-3 - 0 Divide and simplify. See Eample 7.. 7. 9. 0... 3... 6. a + b ab, 6 + + 0, a - b a 3 b - 6 + 9 - - 6, - 9-3 + 6, - 8 + 8 + + - 6-6 - 8 a - a - 6 a - 8 3-3 - 7, - 3 3-7, 8b + 3a + 6 6. 8., + 0 + 6 + 6 + 6, a - 7a - 8 a + 36 + 3 + 9 + 6 + 8, ab - b + 3a - 6 a - a + a - b a 3 + a b + a + b, 6a MIXED a 3 + a 7 3, - 7 8-9 6a b a -, 3ab a - Perform each indicated operation. See Eamples through 8. 7. 8. 9. - 9-9 - - 6-6 + 9-6 + 9 - + 6 - - 30-0 - 7, - 8 + - 6 + 9 60. a + 36 a - 7a - 8, a - a - 6 a - 8 6. Simplify: r3 + s 3 r + s 6. Simplify: m3 - n 3 m - n 63., 3y 6 6. 6. 66. 67. 68. 69. 70. 3y, 6 3 - - y + y - + - 3y - y - y + y - + 7 +, - 9-8 - - y + y - + -, 8y - 6y + y - 9y - a - 0 3a - a, a3 + a 9a3 + 6a a - 8a a - a a - 0 3a - a, a a3 + a 9a3 + 6a a - 8a a - a b + 3 - - 3-0 - 8 - + - 3 + 6 + 0 Find each function value. See Eample 9. 7. If f + 8, find f, f 0, and f -. - 7. If f -, find f -, f 0, and f 0. - + 73. If g + 8 3, find g3, g -, and g - 7. If st t 3 + t, find s -, s, and s. + 7. The total revenue from the sale of a popular book is approimated by the rational function R 000, where is + the number of years since publication and R is the total revenue in millions of dollars. Find the total revenue at the end of the first year. Find the total revenue at the end of the second year. c. Find the revenue during the second year only. d. Find the domain of function R. 76. The function f 00,000 models the cost in dollars for 00 - removing percent of the pollutants from a bayou in which a nearby company dumped creosol. Find the cost of removing 0% of the pollutants from the bayou. [Hint: Find f 0.] Find the cost of removing 60% of the pollutants and then 80% of the pollutants. c. Find f 90, then f 9, and then f 99. What happens to the cost as approaches 00%? d. Find the domain of function f.
Section 6. Rational Functions and Multiplying and Dividing Rational Epressions 37 REVIEW AND PREVIEW Perform each indicated operation. See Section.3. 77. 79. 8. + 3 8-3 8 + - 3 6 78. 80. 8. 0-7 0 3 + 7 9-6 + 3 CONCEPT EXTENSIONS Solve. For Eercises 83 and 8, see the second Concept Check in this section; for Eercises 8 and 86, see the third Concept Check. 83. Which of the epressions are equivalent to -? - - - - + - c. d. - - 8. Which of the epressions are equivalent to - +? - - c. - 8. Does - - d. - - + simplify to? Why or why not? 86. Does + 7 simplify to 7? Why or why not? 87. Find the area of the rectangle. 88. Find the area of the triangle. m m 90. A lottery prize of 3 dollars is to be divided among y people. Epress the amount of money each person is to receive as a rational epression in and y. 9. In your own words, eplain how to simplify a rational epression. 9. In your own words, eplain the difference between multiplying rational epressions and dividing rational epressions. 93. Decide whether each rational epression equals, -, or neither. + + c. + - - e. - + - - d. - - + f. - + - 9. Find the polynomial in the second numerator such that the following statement is true. -? - 7 + 0 + + 9. In our definition of division for P Q, R S we stated that Q 0, S 0, and R 0. Eplain why R cannot equal 0. 96. In your own words, eplain how to find the domain of a rational function. 0 97. Graph a portion of the function f. To do so, 00 - complete the given table, plot the points, and then connect the plotted points with a smooth curve. 0 0 30 0 70 90 9 99 y or f() in. 0y y 6 in. 89. A parallelogram has an area of + - 3 square feet and a height of feet. Epress the length of its base as a - rational epression in. (Hint: Since A b A h, b h or b A, h.) 98. The domain of the function f is all real numbers ecept 0. This means that the graph of this function will be in two pieces: one piece corresponding to values less than 0 and one piece corresponding to values greater than 0. Graph the function by completing the following tables, separately plotting the points, and connecting each set of plotted points with a smooth curve. y or f() h - - - - - b y or f()
38 CHAPTER 6 Rational Epressions Simplify. Assume that no denominator is 0. 99. p - - p 00. 3 + q n q n + 3 n + k - 9 0. n 0. - 6 3 + k Perform the indicated operation. Write all answers in lowest terms. 03. 0. n - 7 3 n - n + n + - 3 8-6 n + 0. yn + 9 0y 06. yn - 6 y n + 07. yn - y n - y n - y n - 3 y n - 8 6y y n + 08. yn + 7y n + 0 0, yn - + y n, yn + y n + y n + 6. Adding and Subtracting Rational Epressions S Add or Subtract Rational Epressions with a Common Denominator. Identify the Least Common Denominator (LCD) of Two or More Rational Epressions. 3 Add or Subtract Rational Epressions with Unlike Denominators. Adding or Subtracting Rational Epressions with a Common Denominator Rational epressions, like rational numbers, can be added or subtracted. We add or subtract rational epressions in the same way that we add or subtract rational numbers (fractions). Adding or Subtracting Rational Epressions with a Common Denominator If P Q and R are rational epressions, then Q P Q + R Q P + R Q and P Q - R Q P - R Q To add or subtract rational epressions with a common denominator, add or subtract the numerators and write the sum or difference over the common denominator. Helpful Hint Very Important: Be sure to insert parentheses here so that the entire numerator is subtracted. EXAMPLE + Add or subtract. 7z + 7z c. + 7-9 + 7 d. 3y - + 3y The rational epressions have common denominators, so add or subtract their numerators and place the sum or difference over their common denominator. + + 6 3 c. d. 7z + 7z + 7z + 7-9 + 7-9 + 7 + 7-7 + 7 3y - + 3y - 7 Simplify. Add the numerators and write the result over the common denominator. Subtract the numerators and write the result over the common denominator. Factor the numerator. - + 3y Subtract the numerators. - - 3y Use the distributive property. - 3y Simplify.