CH 39 USING THE GCF TO REDUCE FRACTIONS

Transcription

1 359 CH 39 USING THE GCF TO EDUCE FACTIONS educig Algeric Frctios M ost of us lered to reduce rithmetic frctio dividig the top d the ottom of the frctio the sme (o-zero) umer. For exmple, Whe it comes to lger, though, we eed to look t reducig frctio little differetl, sice it s kid of hrd to divide letters letters. So ow we re-exmie the reducig of rithmetic frctios so tht we get method more pproprite to lgeric frctios. Wtch this: which is the sme swer s efor Here s wht we did. First we fctored the top d the ottom ito prime fctors -- the ver uildig locks of the whole umers. The we divided out fctor tht ws commo to oth the top d the ottom (sice umer divided itself is ). Whtever fctors remi costitute the fil swer., EXAMPLE : educe ech frctio: A

2 360 Notice tht we do t leve zero i the umertor just ecuse ever fctor i the umertor cceled out. ememer tht ccelig is rell dividig, so ech time we ccel pir of fctors, we re rell dividig umer itself, which is lws. B. Becuse d re eig multiplied (the re oe term), we c divide top d ottom. C. c c cw cw w The top d ottom ech cosist oe term, so we c divide top d ottom c. EXAMPLE : educe: + + c Solutio: C we divide top d ottom? I do t thik so. But me ou thik so; give me momet to expli. Sice dividig is desiged to remove multiplictio, d sice the top d ottom hve dditio i them (the re ech two terms), we cot ccel the s. Therefore, this frctio is ot reducil Here s more cocrete w to verif this fct. If ou thik ou should e le to ccel the s i this frctio, let s do it d see wht hppes whe we use umers. Suppose 6, 0, d c. Usig the Order of Opertios, we would get c O the other hd, ccelig the s completel would produce the followig outcome: 0 5 c c 6

3 36 Ad if ou thik tht it would e legl to divide out the s d leve s i their plce, the followig would e the cosequece: 0 c c 3 Tht s three differet swers, depedig upo our theor of reducig frctios. Which oe s right? Well, sice the Order of Opertios hs ee with us for log time ow, I prefer to stick with wht I kow, d coclude tht the ol vlid swer is 8. I therefore thik we should reject ll this goof ccelig; 9 oe of it worked. The ol time we c ccel i frctio is whe oth the top d the ottom cosist of oe term (i., the fil opertio is multiplictio). Sice reducig frctio requires oe term o the top d oe term o the ottom, we c ow utilize our kowledge of fctorig to reduce frctios which do t iitill possess this propert. EXAMPLE 3: educe ech frctio: A. x 8 ( x 4) ( x 4) 0 ( 5) ( 5) x 4 5 B. x x x( ) x ( ) x xz x( z) x ( z) z C. D. ( ) ( ) ( ) q q q( ) q( ) q( ) ux uw u( x w) x w ( x w ) q There s o commo fctor to ccel. So the origil frctio is ot reducil

4 36 EXAMPLE 4: educe ech frctio: A. B. x x x( x ) x ( x ) x x x ( ) ( ) x C. D. 3 ( 3) ( 3) u 4 ( u 4) ( u 4) u 4u uu ( 4) uu ( 4) 3 u We ow hve techique for reducig frctios to lowest terms, ut it m still e little hrd to elieve, for istce, tht, s i prt B of the previous exmpl Perhps ou ll feel little more cofidet i this swer if we sustitute umer for d verif the equlit ourselves. So let s choose 0; the , which equls The Steps Needed to educe Frctio:. Be sure the top is fctored.. Be sure the ottom is fctored. 3. Divide top d ottom commo fctors.

5 363 Homework. educe ech frctio fctorig ito primes: c d g i j educe ech frctio:. 3 x 3. x x c. tx tz t tz d. x z T T mx m x g. x x x cx x x i. m j. PT QT T k. g h g h l. c cx x 3. educe ech frctio:. x 3 x x. c. z z z d. Q Q 3 3 g. t t t 9 i. 3 3 j. c c c k. z 3z z 3 l. Q 0Q Q m. c c c. rx x r o. x x x xz p.

6 364 Prctice Prolems 4. educe ech frctio:. 5x 0 0x 45. x x cx dx c. m d. w w 3w t t t 4 g. x 8 3x x x 5. educe ech frctio:. 5x 0 0x 45. c d c. cm c c d. u u 7u t t t 99 g. x 8 6x x x Solutios c. 4 9 d g. 3 3 i. 5 j. 4.. x 4 7 i. m. x c. x z d. x z z Not reducile g. x c j. P Q k. g h l. Not reducile g h

7 x + 3. c. g. t i. m. c z d. 9 j. Not reducile k. z l. Q 0. Not reducile o. x z p. Not reducile 4.. x 4. c. m + d. x 9 c d t 4 g. Not reducile w x x 9 t. c d + 99 g. c. m + d. x 4 3x 6 u 7 Not reducile

8 366 Eductio is ot received. It is chieved. Aomous