Section A4 Rational Expressions: Basic Operations


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1 A Appendi A A BASIC ALGEBRA REVIEW 7. Construction. A rectngulr opentopped bo is to be constructed out of 9 by 6inch sheets of thin crdbord by cutting inch squres out of ech corner nd bending the sides up. Epress ech of the following quntities s polynomil in both fctored nd epnded form. The re of crdbord fter the corners hve been removed. (B) The volume of the bo. Section A Rtionl Epressions: Bsic Opertions Reducing to Lowest Terms Multipliction nd Division Addition nd Subtrction Compound Frctions We now turn our ttention to frctionl forms. A quotient of two lgebric epressions, division by 0 ecluded, is clled frctionl epression. If both the numertor nd denomintor of frctionl epression re polynomils, the frctionl epression is clled rtionl epression. Some emples of rtionl epressions re the following (recll, nonzero constnt is polynomil of degree 0): In this section we discuss bsic opertions on rtionl epressions, including multipliction, division, ddition, nd subtrction. Since vribles represent rel numbers in the rtionl epressions we re going to consider, the properties of rel number frctions summrized in Section A ply centrl role in much of the work tht we will do. Even though not lwys eplicitly stted, we lwys ssume tht vribles re restricted so tht division by 0 is ecluded. Reducing to Lowest Terms We strt this discussion by restting the fundmentl property of frctions (from Theorem in Section A): FUNDAMENTAL PROPERTY OF FRACTIONS If, b, nd k re rel numbers with b, k 0, then k kb b ( ) ( ) 0, Using this property from left to right to eliminte ll common fctors from the numertor nd the denomintor of given frction is referred to s reducing
2 A Rtionl Epressions: Bsic Opertions A frction to lowest terms. We re ctully dividing the numertor nd denomintor by the sme nonzero common fctor. Using the property from right to left tht is, multiplying the numertor nd the denomintor by the sme nonzero fctor is referred to s rising frction to higher terms. We will use the property in both directions in the mteril tht follows. We sy tht rtionl epression is reduced to lowest terms if the numertor nd denomintor do not hve ny fctors in common. Unless stted to the contrry, fctors will be reltive to the integers. Reducing Rtionl Epressions Reduce ech rtionl epression to the lowest terms. (B) ( )( ) ( )( ) ( ) ( )( ) Fctor numertor nd denomintor completely. Divide numertor nd denomintor by ( ); this is vlid opertion s long s nd. Dividing numertor nd denomintor by ( ) cn be indicted by drwing lines through both ( )s nd writing the resulting quotients, s. nd Reduce ech rtionl epression to lowest terms. 6 (B) 8 8 Reducing Rtionl Epression Reduce the following rtionl epression to lowest terms. 6 ( ) ( ) ( ) [ ( )] 8 8 ( ) ( ) 8 ( ) ( )( )
3 A Appendi A A BASIC ALGEBRA REVIEW Reduce the following rtionl epression to lowest terms. 6 ( ) ( ) 6 CAUTION Remember to lwys fctor the numertor nd denomintor first, then divide out ny common fctors. Do not indiscrimintely eliminte terms tht pper in both the numertor nd the denomintor. For emple, y y y y Since the term y is not fctor of the numertor, it cnnot be eliminted. In fct, ( y )/y is lredy reduced to lowest terms. Multipliction nd Division Since we re restricting vrible replcements to rel numbers, multipliction nd division of rtionl epressions follow the rules for multiplying nd dividing rel number frctions (Theorem in Section A). MULTIPLICATION AND DIVISION If, b, c, nd d re rel numbers with b, d 0, then:. b c d c bd. b c d b d c c 0 ( ) Eplore/Discuss Write verbl description of the process of multiplying two frctions. Do the sme for the quotient of two frctions.
4 A Rtionl Epressions: Bsic Opertions A Multiplying nd Dividing Rtionl Epressions Perform the indicted opertions nd reduce to lowest terms. 0 y y 9y 9 0 y ( )( ) y( ) ( ) 6 Fctor numertors nd denomintors; then divide ny numertor nd ny denomintor with like common fctor. ( ) (B) ( ) is the sme s. ( ) b ( b), useful chnge in ( ) ( ) some problems. (C) y y y y y y y ( y y ) y( y)( y) ( y) ( y)( y y ) y y( y) Perform the indicted opertions nd reduce to lowest terms. (C) y y 6y y 6y 9 y 9y (B) m n m mn n m n m n mn m n m n ( ) 6
5 A6 Appendi A A BASIC ALGEBRA REVIEW Addition nd Subtrction Agin, becuse we re restricting vrible replcements to rel numbers, ddition nd subtrction of rtionl epressions follow the rules for dding nd subtrcting rel number frctions (Theorem in Section A). ADDITION AND SUBTRACTION For, b, nd c rel numbers with b 0:.. b c b c b b c b c b y ( ) y y Thus, we dd rtionl epressions with the sme denomintors by dding or subtrcting their numertors nd plcing the result over the common denomintor. If the denomintors re not the sme, we rise the frctions to higher terms, using the fundmentl property of frctions to obtin common denomintors, nd then proceed s described. Even though ny common denomintor will do, our work will be simplified if the lest common denomintor (LCD) is used. Often, the LCD is obvious, but if it is not, the steps in the bo describe how to find it. THE LEAST COMMON DENOMINATOR (LCD) The LCD of two or more rtionl epressions is found s follows:. Fctor ech denomintor completely.. Identify ech different prime fctor from ll the denomintors.. Form product using ech different fctor to the highest power tht occurs in ny one denomintor. This product is the LCD. Adding nd Subtrcting Rtionl Epressions Combine into single frction nd reduce to lowest terms. (C) 0 6 (B) 9 6y Solutions To find the LCD, fctor ech denomintor completely: 0 6 LCD 90
6 A Rtionl Epressions: Bsic Opertions A7 Now use the fundmentl property of frctions to mke ech denomintor 90: (B) LCD y 8y 6y y 8y 8y 8y y y y 9 6y 8y 8y (C) ( ) ( )( ) Note: ( ) We hve gin used the fct tht b (b ). The LCD ( ) ( ). Thus, ( ) ( )( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 6 9) ( 6) ( 9) ( ) ( ) ( ) ( ) 7 0 ( ) ( ) Be creful of sign errors here. Combine into single frction nd reduce to lowest terms. (C) (B) y y y y y y
7 A8 Appendi A A BASIC ALGEBRA REVIEW Eplore/Discuss 6 Wht is the vlue of? Wht is the result of entering 6 on clcultor? Wht is the difference between 6 ( ) nd (6 )? How could you use frction brs to distinguish between these two cses 6 when writing? Compound Frctions A frctionl epression with frctions in its numertor, denomintor, or both is clled compound frction. It is often necessry to represent compound frction s simple frction tht is (in ll cses we will consider), s the quotient of two polynomils. The process does not involve ny new concepts. It is mtter of pplying old concepts nd processes in the right sequence. We will illustrte two pproches to the problem, ech with its own merits, depending on the prticulr problem under considertion. Simplifying Compound Frctions Epress s simple frction reduced to lowest terms. Solution Method. Multiply the numertor nd denomintor by the LCD of ll frctions in the numertor nd denomintor in this cse,. (We re multiplying by / ). ( ) ( )( )
8 A Rtionl Epressions: Bsic Opertions A9 Method. Write the numertor nd denomintor s single frctions. Then tret s quotient. ( )( ) Epress s simple frction reduced to lowest terms. Use the two methods described in Emple. 6 Simplifying Compound Frctions Epress s simple frction reduced to lowest terms. y y y y Solution Using the first method described in Emple, we hve y y y y y y y y y y y y y y y y y (y )(y y ) y(y )(y ) y y y(y ) 6 Epress s simple frction reduced to lowest terms. Use the first method described in Emple. b b b b
9 A0 Appendi A A BASIC ALGEBRA REVIEW Answers to Mtched Problems ( ) ( )( ). (B).. (B) (C) mn y 9y 6 b. (B) (C). 6. (y ) (y ) b EXERCISE A A In Problems 0, perform the indicted opertions nd reduce nswers to lowest terms. Represent ny compound frctions s simple frctions reduced to lowest terms.. d. d 6 d y y ( ) m m. m m n m n m mn m mn n b b b y y y y 9 y y y d d 6 y y d m 8m m m 6m 9 ( ) B Problems 6 re clculusrelted. Reduce ech frction to lowest terms In Problems 7 0, perform the indicted opertions nd reduce nswers to lowest terms. Represent ny compound frctions s simple frctions reduced to lowest terms ( ) ( ) ( ) ( )( ) y y y y y y 8y m m m 6 m m 7 b y 9 by y c c c c c c y y ( ) ( ) 6 ( ) 9 ( ) ( ) 6 ( ) 8 ( ) ( ) 8 ( ) ( )( ) ( ) 6 ( ) 6 y y y y y
10 Problems re clculusrelted. Perform the indicted opertions nd reduce nswers to lowest terms. Represent ny compound frctions s simple frctions reduced to lowest terms. h.. h ( h) h h h.. h h In Problems, imgine tht the indicted solutions were given to you by student whom you were tutoring in this clss. Is the solution correct? If the solution is incorrect, eplin wht is wrong nd how it cn be corrected. (B) Show correct solution for ech incorrect solution y y y y y y y y y y 6 y y y y ( h) ( h) ( ) h h C A Integer Eponents A y. y y y y y y y ( h) 8. ( ) y y h In Problems 6, perform the indicted opertions nd reduce nswers to lowest terms. Represent ny compound frctions s simple frctions reduced to lowest terms. y y y.. y. 6. In Problems 7 nd 8,, b, c, nd d represent rel numbers. 7. Prove tht d/c is the multiplictive inverse of c/d (c, d 0). (B) Use prt A to prove tht 8. Prove tht b c d b d c b c b c b b 0 s s t s t s t t b, c, d 0 Section A Integer Eponents Integer Eponents Scientific Nottion The French philosopher/mthemticin René Descrtes (96 60) is generlly credited with the introduction of the very useful eponent nottion n. This
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