1.4 RATIONAL EXPRESSIONS

Transcription

1 6 CHAPTER Fundamentals.4 RATIONAL EXPRESSIONS Te Domain of an Algebraic Epression Simplifying Rational Epressions Multiplying and Dividing Rational Epressions Adding and Subtracting Rational Epressions Compound Fractions Rationalizing te Denominator or te Numerator Avoiding Common Errors A quotient of two algebraic epressions is called a fractional epression. Here are some eamples: y y " A rational epression is a fractional epression in wic bot te numerator and te denominator are polynomials. For eample, te first tree epressions in te above list are rational epressions, but te fourt is not, since its denominator contains a radical. In tis section we learn ow to perform algebraic operations on rational epressions. Te Domain of an Algebraic Epression Epression Domain 5 0? 06! ! In general, an algebraic epression may not be defined for all values of te variable. Te domain of an algebraic epression is te set of real numbers tat te variable is permitted to ave. Te table in te margin gives some basic epressions and teir domains. EXAMPLE Finding te Domain of an Epression Find te domains of te following epressions. (a) (b) 5 6 (c)! 5 (a) Tis polynomial is defined for every. Tus te domain is te set of real numbers. (b) We first factor te denominator. 5 6 Denominator would be 0 if or Since te denominator is zero wen or, te epression is not defined for tese numbers. Te domain is 5 0? and? 6. (c) For te numerator to be defined, we must ave 0. Also, we cannot divide by zero, so? 5. Must ave 0 to take square root! 5 Denominator would be 0 if 5 Tus te domain is and? 56. Now Try Eercise Copyrigt 06 Cengage Learning. All Rigts Reserved. May not be copied, scanned, or duplicated, in wole or in part. Due to electronic rigts, some tird party content may be suppressed from te ebook and/or ecapter(s). Editorial review as deemed tat any suppressed content does not materially affect te overall learning eperience. Cengage Learning reserves te rigt to remove additional content at any time if subsequent rigts restrictions require it.

2 SECTION.4 Rational Epressions 7 Simplifying Rational Epressions To simplify rational epressions, we factor bot numerator and denominator and use te following property of fractions: AC BC A B Tis allows us to cancel common factors from te numerator and denominator. EXAMPLE Simplifying Rational Epressions by Cancellation Simplify: We can t cancel te s in because is not a factor. Now Try Eercise 9 Cancel common factors Multiplying and Dividing Rational Epressions To multiply rational epressions, we use te following property of fractions: A # C B D AC BD Tis says tat to multiply two fractions, we multiply teir numerators and multiply teir denominators. EXAMPLE Multiplying Rational Epressions Perform te indicated multiplication and simplify: We first factor. 4 # # Now Try Eercise 7 # Property of fractions 4 Cancel common factors Copyrigt 06 Cengage Learning. All Rigts Reserved. May not be copied, scanned, or duplicated, in wole or in part. Due to electronic rigts, some tird party content may be suppressed from te ebook and/or ecapter(s). Editorial review as deemed tat any suppressed content does not materially affect te overall learning eperience. Cengage Learning reserves te rigt to remove additional content at any time if subsequent rigts restrictions require it.

3 8 CHAPTER Fundamentals To divide rational epressions, we use te following property of fractions: A B 4 C D A B # D C Tis says tat to divide a fraction by anoter fraction, we invert te divisor and multiply. EXAMPLE 4 Dividing Rational Epressions Perform te indicated division and simplify: # Invert and multiply Now Try Eercise Cancel common factors Avoid making te following error: A B C A B A C For instance, if we let A, B, and C, ten we see te error: Wrong! Adding and Subtracting Rational Epressions To add or subtract rational epressions, we first find a common denominator and ten use te following property of fractions: A C B C A B C Altoug any common denominator will work, it is best to use te least common denominator (LCD) as eplained in Section.. Te LCD is found by factoring eac denominator and taking te product of te distinct factors, using te igest power tat appears in any of te factors. EXAMPLE 5 Adding and Subtracting Rational Epressions Perform te indicated operations and simplify. (a) (b) (a) Here te LCD is simply te product. 6 6 Write fractions using LCD Add fractions Combine terms in numerator Copyrigt 06 Cengage Learning. All Rigts Reserved. May not be copied, scanned, or duplicated, in wole or in part. Due to electronic rigts, some tird party content may be suppressed from te ebook and/or ecapter(s). Editorial review as deemed tat any suppressed content does not materially affect te overall learning eperience. Cengage Learning reserves te rigt to remove additional content at any time if subsequent rigts restrictions require it.

4 SECTION.4 Rational Epressions 9 (b) Te LCD of and is. Combine fractions using LCD Distributive Property Now Try Eercises 4 and 45 Combine terms in numerator Compound Fractions A compound fraction is a fraction in wic te numerator, te denominator, or bot, are temselves fractional epressions. EXAMPLE 6 Simplifying a Compound Fraction Simplify: y y We combine te terms in te numerator into a single fraction. We do te same in te denominator. Ten we invert and multiply. y y y y y # y y y y y y Matematics in te Modern World Courtesy of NASA Error-Correcting Codes Te pictures sent back by te Patfinder spacecraft from te surface of Mars on July 4, 997, were astoundingly clear. But few viewing tese pictures were aware of te comple matematics used to accomplis tat feat. Te distance to Mars is enormous, and te background noise (or static) is many times stronger tan te original signal emitted by te spacecraft. So wen scientists receive te signal, it is full of errors. To get a clear picture, te errors must be found and corrected. Tis same problem of errors is routinely encountered in transmitting bank records wen you use an ATM macine or voice wen you are talking on te telepone. To understand ow errors are found and corrected, we must first understand tat to transmit pictures, sound, or tet, we transform tem into bits (te digits 0 or ; see page 8). To elp te receiver recognize errors, te message is coded by inserting additional bits. For eample, suppose you want to transmit te message 000. A very simpleminded code is as follows: Send eac digit a million times. Te person receiving te message reads it in blocks of a million digits. If te first block is mostly s, e concludes tat you are probably trying to transmit a, and so on. To say tat tis code is not efficient is a bit of an understatement; it requires sending a million times more data tan te original message. Anoter metod inserts ceck digits. For eample, for eac block of eigt digits insert a nint digit; te inserted digit is 0 if tere is an even number of s in te block and if tere is an odd number. So if a single digit is wrong (a 0 canged to a or vice versa), te ceck digits allow us to recognize tat an error as occurred. Tis metod does not tell us were te error is, so we can t correct it. Modern error-correcting codes use interesting matematical algoritms tat require inserting relatively few digits but tat allow te receiver to not only recognize, but also correct, errors. Te first error-correcting code was developed in te 940s by Ricard Hamming at MIT. It is interesting to note tat te Englis language as a built-in error correcting mecanism; to test it, try reading tis error-laden sentence: Gve mo libty o giv ne det. Copyrigt 06 Cengage Learning. All Rigts Reserved. May not be copied, scanned, or duplicated, in wole or in part. Due to electronic rigts, some tird party content may be suppressed from te ebook and/or ecapter(s). Editorial review as deemed tat any suppressed content does not materially affect te overall learning eperience. Cengage Learning reserves te rigt to remove additional content at any time if subsequent rigts restrictions require it.

5 40 CHAPTER Fundamentals We find te LCD of all te fractions in te epression, ten multiply numerator and denominator by it. In tis eample te LCD of all te fractions is y. Tus y y y # y y y y y y Simplify Multiply numerator and denominator by y y y y Now Try Eercises 59 and 65 Te net two eamples sow situations in calculus tat require te ability to work wit fractional epressions. We can also simplify by multiplying te numerator and te denominator by aa. EXAMPLE 7 Simplifying a Compound Fraction Simplify: a a We begin by combining te fractions in te numerator using a common denominator. a a Now Try Eercise 7 a a aa a a # aa a a aa # aa # aa Combine fractions in te numerator Property of fractions (invert divisor and multiply) Distributive Property Simplify Property 5 of fractions (cancel common factors) out te power of wit te smallest eponent, in tis case /. EXAMPLE 8 Simplifying a Compound Fraction Simplify: / / / from te numerator. / / / 4 / / Copyrigt 06 Cengage Learning. All Rigts Reserved. May not be copied, scanned, or duplicated, in wole or in part. Due to electronic rigts, some tird party content may be suppressed from te ebook and/or ecapter(s). Editorial review as deemed tat any suppressed content does not materially affect te overall learning eperience. Cengage Learning reserves te rigt to remove additional content at any time if subsequent rigts restrictions require it.

6 SECTION.4 Rational Epressions 4 Since / / / is a fraction, we can clear all fractions by multiplying numerator and denominator by /. / / Now Try Eercise 8 / / # / / / / Rationalizing te Denominator or te Numerator If a fraction as a denominator of te form A B!C, we can rationalize te denominator by multiplying numerator and denominator by te conjugate radical A B!C. Tis works because, by Special Product Formula in Section., te product of te denominator and its conjugate radical does not contain a radical: A B!C A B!C A B C Special Product Formula A BA B A B EXAMPLE 9 Rationalizing te Denominator Rationalize te denominator:! We multiply bot te numerator and te denominator by te conjugate radical of!, wic is!. Now Try Eercise 85! #!!! Multiply numerator and denominator by te conjugate radical! Special Product Formula!!!! Special Product Formula A BA B A B EXAMPLE 0 Rationalizing te Numerator Rationalize te numerator:!4 We multiply numerator and denominator by te conjugate radical!4.!4!4!4!4 Now Try Eercise 9 4 4!4 #!4!4!4!4 Multiply numerator and denominator by te conjugate radical Special Product Formula Property 5 of fractions (cancel common factors) Copyrigt 06 Cengage Learning. All Rigts Reserved. May not be copied, scanned, or duplicated, in wole or in part. Due to electronic rigts, some tird party content may be suppressed from te ebook and/or ecapter(s). Editorial review as deemed tat any suppressed content does not materially affect te overall learning eperience. Cengage Learning reserves te rigt to remove additional content at any time if subsequent rigts restrictions require it.

7 4 CHAPTER Fundamentals Avoiding Common Errors Don t make te mistake of applying properties of multiplication to te operation of addition. Many of te common errors in algebra involve doing just tat. Te following table states several properties of multiplication and illustrates te error in applying tem to addition. Correct multiplication property Common error wit addition a # b a # b a b a b!a # b!a!b a, b 0!a b!a!b "a # b a # b a, b 0 "a b a b # a b a # b a b a b ab a b a # b a # b a b a b a b a b To verify tat te equations in te rigt-and column are wrong, simply substitute numbers for a and b and calculate eac side. For eample, if we take a and b in te fourt error, we get different values for te left- and rigt-and sides: a b a b 4 Left-and side Rigt-and side Since? 4, te stated equation is wrong. You sould similarly convince yourself of te error in eac of te oter equations. (See Eercises 0 and 0.).4 EXERCISES CONCEPTS. Wic of te following are rational epressions? (a) (b)! (c). To simplify a rational epression, we cancel factors tat are common to te and. So te epression simplifies to.. To multiply two rational epressions, we multiply teir So togeter and multiply teir # is te same as. 4. Consider te epression. togeter. (a) How many terms does tis epression ave? (b) Find te least common denominator of all te terms. (c) Perform te addition and simplify. 5 6 Yes or No? If No, give a reason. (Disregard any value tat makes a denominator zero.) 5. (a) Is te epression equal to? (b) Is te epression " 5 equal to 5? 6. (a) Is te epression a equal to a? (b) Is te epression 4 equal to? SKILLS 7 4 Domain Find te domain of te epression !.. 0. t 5 t 6 4.!! Copyrigt 06 Cengage Learning. All Rigts Reserved. May not be copied, scanned, or duplicated, in wole or in part. Due to electronic rigts, some tird party content may be suppressed from te ebook and/or ecapter(s). Editorial review as deemed tat any suppressed content does not materially affect te overall learning eperience. Cengage Learning reserves te rigt to remove additional content at any time if subsequent rigts restrictions require it.

8 SECTION.4 Rational Epressions Simplify Simplify te rational epression y y y y y 8 y 7y Multiply or Divide Perform te multiplication or division and simplify # t # t t 9 t 9 # 5. 7 # y y # y y y y y /y z # # y/z # 9 58 Add or Subtract Perform te addition or subtraction and simplify u u u 48. a ab 4 b Compound Fractions Simplify te compound fractional epression y y y y y y 69. y y 7. y 60. y c c 4 y y y 68. y y y 70. y y Epressions Found in Calculus Simplify te fractional epression. (Epressions like tese arise in calculus.) !! Copyrigt 06 Cengage Learning. All Rigts Reserved. May not be copied, scanned, or duplicated, in wole or in part. Due to electronic rigts, some tird party content may be suppressed from te ebook and/or ecapter(s). Editorial review as deemed tat any suppressed content does not materially affect te overall learning eperience. Cengage Learning reserves te rigt to remove additional content at any time if subsequent rigts restrictions require it.

9 44 CHAPTER Fundamentals a Å " b 78. Å a 4 b Epressions Found in Calculus Simplify te epression. (Tis type of epression arises in calculus wen using te quotient rule. ) / / 8. / / 8. / / / / 7 / Rationalize Denominator Rationalize te denominator !!!7 y!!y !5! y!!y 9 96 Rationalize Numerator Rationalize te numerator. 9.!5 9.!r! !!5!!!! 95. " 96.!! APPLICATIONS 97. Electrical Resistance If two electrical resistors wit resistances R and R are connected in parallel (see te figure), ten te total resistance R is given by R R R (a) Simplify te epression for R. (b) If R 0 oms and R 0 oms, wat is te total resistance R? R R 98. Average Cost A cloting manufacturer finds tat te cost of producing sirts is dollars. (a) Eplain wy te average cost per sirt is given by te rational epression A (b) Complete te table by calculating te average cost per sirt for te given values of Average cost DISCUSS DISCOVER PROVE WRITE 99. DISCOVER: Limiting Beavior of a Rational Epression Te rational epression 9 is not defined for. Complete te tables, and determine wat value te epression approaces as gets closer and closer to. Wy is tis reasonable? te numerator of te epression and simplify to see wy DISCUSS WRITE: Is Tis Rationalization? In te epression /! we would eliminate te radical if we were to square bot numerator and denominator. Is tis te same ting as rationalizing te denominator? Eplain. Copyrigt 06 Cengage Learning. All Rigts Reserved. May not be copied, scanned, or duplicated, in wole or in part. Due to electronic rigts, some tird party content may be suppressed from te ebook and/or ecapter(s). Editorial review as deemed tat any suppressed content does not materially affect te overall learning eperience. Cengage Learning reserves te rigt to remove additional content at any time if subsequent rigts restrictions require it.