Appendix A Examples for Labs 1, 2, 3 1. FACTORING POLYNOMIALS
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1 Appedi A Emples for Ls,,. FACTORING POLYNOMIALS Tere re m stdrd metods of fctorig tt ou ve lered i previous courses. You will uild o tese fctorig metods i our preclculus course to ele ou to fctor epressios tt will ot fctor trditiol metods. First ou eed to mke sure ou re skilled t te stdrd metods of fctorig. A. Gretest Commo Fctors Emple. Fctor 8 Sice ec term cotis fctor So 8 ) Emple. Fctor ) ) Te commo fctor m coti iomil term. Ec term cotis fctor ) So ) ) ) B. Differece of Squres B, fctor out tis GCF [ ) ] ) ) ) ) A A B) A B) Emple. Fctor 9t 8v Tis oe is esil recogizle s differece of two perfect squres. So 9 8v 7t 9v 7t 9v t ) ) 8 Emple. Fctor Mke sure ou completel fctor te epressio, rememerig tt te sum of squres does ot fctor. 8 ) ) ) ) ) C. Sum or Differece of Cues MEMORIZE! A B A B) A AB B ) Formuls A B A B) A AB B ) Emple. Fctor 8 Use te formul for sum of cues. 8 Emple. Fctor m 7 Use te formul for differece of cues. 7 m ) m m 9 m ) ) 0 )
2 D. Triomils Tere re m metods of fctorig geerl triomils; usull guess d ceck metod is efficiet. Tik of te fctorig s workig ckwrds from multiplictio Fctor B Guess d Ceck: ) ) Ceck multiplictio: ) ) 9 8. Groupig strtegies If epressio s four or more terms, we tr groupig strtegies to tr to fctor te epressio. Emple : Fctor m m 9 p Tik of tis emple s differece of two perfect squres: Groupig te terms first:: m m 9 p m m 9) p m ) p m p) m p) Emple : Fctor Te polomil m coti iomil commo fctor: Groupig te terms i pirs: ) ). RATIONAL EXPRESSIONS ) ) ) ) ) ) ) Rememer te properties of frctios, listed elow to elp ou recll tem. It is importt to rememer tt divisio zero is udefied over te rel umers, so o frctio c ve deomitor of zero. If,,c d d re rel umers d o deomitor is zero, te: c if d ol if d c d c d c d c d d c c c c c Skills i workig wit rtiol epressios re importt to success i our preclculus course. You will eed to e le to dd, sutrct, multipl d divide rtiol
3 epressios. Tis icludes simplifig comple frctios. Te emples elow will illustrte te procedures for simplifig rtiol epressios. A. Simplifictio, multiplictio, divisio Alws FACTOR first! NOTE: Ol commo fctors ccel out, NOT terms! Tis is ecuse ccelig is rell multiplictio. Emple. 0 ) ) fctor out - to see te commo fctor) Emple Emple mp m p ) ) ) ) ) ) ) ) do t ccel more!) ) 0mp m p 8 ) multipl reciprocl) 9m p B. Additio d sutrctio To dd or sutrct frctios, ou must first fid commo deomitor, idell te lest c c commo deomitor, d te use te propert. Emple : ) ) ) ) Write frctio wit lcd ) ) ) ) ) ) ) ) Add d sutrct ) ) Simplif ) ) Collect terms ) ) Reduce to lowest terms if ecessr. Tis emple is lred i lowest terms.
4 C. Comple Frctios Oe metod of simplifig comple frctio is te followig: simplif te umertor d te deomitor to sigle frctios, d te divide. A comple frctio is rell divisio questio. Emple: Write s quotiet of frctios Divisio prolem Multipl reciprocl ) ) ) Fctor d ccel ). POLYNOMIAL MULTIPLICATION AND SPECIAL PRODUCTS Multiplig polomils ofte ivolves te specil products sow elow. You ve used tese ptters we fctorig ) ) ) ) Emple: ) ) ) 9 ) 8 ) ALGEBRAIC LONG DIVISION I erlier emples, fctorig ws used to do polomil divisio. We te divisio cot e completed fctorig, lgeric log divisio c e used. Dividig polomils lgericll uses te sme procedure lgoritm) we use i ritmetic. Te emples elow will illustrte te procedures for divisio.
5 Emple : Rememer log divisio wit wole umers Te quotiet is d te remider is. We usull write te swer s mied umerl or deciml.. Emple : Dividig polomils fctorig. ) ) ) Emple : Usig log divisio: Te quotiet is 9 9 Emple : Quotiet otice te zero terms 8 Aswer: Remider
6 . SOLVING EQUATIONS We solvig equtios lgericll, it is ofte elpful to clssif te tpe of equtio ou re solvig so ou c ppl suitle tecique to tt prticulr tpe of equtio. Te emples elow will elp ou review some commo equtio solvig teciques. Emple : Solvig lier equtio ) ) Simplif ec side Use opposites to collect vriles to os side Simplif d solve Te solutio c e verified sustitutio i te origil equtio. Sice divisio zero is udefied, te vrile cot ve vlue tt would require divisio zero. Solutios must e cecked for equtios wit vriles i te deomitor. Emple : Solve Fctor to fid LCD ) ) ) ) Determie restrictios 0,, ) sice tese vlues would mke te deomitor zero d te frctios would e udefied. ) )) ) )) ) ) ) ) Multipl ec side LCD to cler frctios ) ) ) Solve s i emple 0 0 But 0 so tere is o rel solutio for tis equtio. Ofte polomil equtios c e solved fctorig metod. Te ke to tis metod is wt is referred to s te Zero Product rule, ie.if te product of two or more rel umers is zero, te t lest oe of te fctors is equl to zero. Emple : Solve: 0 Set te equtio equl to 0 ) ) 0 Fctor or Use zero product rule to solve
7 7 Emple : Solve: 9 0 ) 9 ) 0 Fctor groupig ) 9) 0 ) ) ) 0 or or Use zero product rule to solve Te qudrtic formul c e used to solve qudrtic equtio tt will ot fctor, or eve oes tt do fctor. Te formul is derived from te geerl qudrtic equtio c 0 were 0. Te solutios re give ± c. Note tt if c < 0, te rdicl is udefied d te equtio s o solutio. Emple : Solve 0 Sustitute te vlues,, c i te formul ) ± ± 7 7 Te two solutios re ) ) d ) ) 7. We solvig equtios ivolvig rdicls ou must e sure to ceck our solutio. Emple : Solve: Isolte te rdicl ) ) Squre ec side 0 Solve fctorig ) ) 0 or Ceck solutios Ceckig te two solutios sows tt is solutio to te origil equtio ut is ot. Terefore is te ol solutio to te equtio.
8 8. USING INTERVAL NOTATION Itervls, sets of rel umers occurrig etwee specified poits, occur i solutios of iequlities s well s oter lgeric sttemets suc s descriptios of te domi d rge of fuctios). Itervl ottio is efficiet w of descriig itervls. Te tle elow sows emples of itervls. Rememer tt ope ed-poits re desigted preteses ) d closed ed-poits squre rckets [ ] Set ottio Itervl ottio Itervl descried i words :. { },] from egtive ifiit up to d icludig. { < },) from egtive ifiit up to ut ot icludig. { > }, ) from to ifiit ot icludig. { } [, ) from to ifiit icludig. { < } [,) from - to icludig - ut ot. { } [,] from - to icludig ot - d 7. { < < },) from - to ecludig ot - d 8. { < },] from - to ecludig - ut icludig 9. { < or } ) [, ), from egtive ifiit up to ut ot icludig - s well s from to ifiit icludig 0. { or }, ], from egtive ifiit up to d icludig - [ ) Itervl Nottio is muc esier t words! s well s from to ifiit icludig. 7. EXPONENTS Recll te lws of epoets: If m, d p re turl umers, te: Multiplictio m Lw: m Power Lws: ) m m m ) p Divisio Lw: mp p m B defiitio: If is turl umer, te... fctors) If 0, If 0, 0 If 0 d is o-zero iteger, te m m p mp p
9 9 If 0 d is turl umer, te If 0 d is odd turl umer, te If 0 d is eve turl umer, te m mes m or ) m is ot rel umer. NOTE: m. Terefore for / to e rel umer, must first e rel umer. Some emples will elp ou sort out tese defiitios Emple : Sometimes ou m wt to simplif te epressio iside te rcket first. Emple : z z ) z ) z z z 9 0 Sometimes ou m wt to use te power rule first. Emple : ) z ) Eiter procedure will give ou te correct result. Tr ec emple gi, usig te metod of emple for emple d vice vers. /
10 0 8. SIMPLIFYING RADICAL EXPRESSIONS Bsic Properties to rememer: If 0, 0,te, 0 ), 0) Te followig emples illustrte ow to epress rdicl epressio. i simplest form For tese emples we will ssume ll vriles represet o-egtive rel umers. Emple p q 7 p p q r r 7 r p q r 7 pr Emple : Emple : RATIONALIZING DENOMINATORS Rdicls i simplest form sould ot ve rdicl i te deomitor. A procedure to elimite te rdicl i deomitor is clled rtiolizig te deomitor, ie. mkig it rtiol! We eed to multipl te umertor d deomitor rdicl tt will mke te deomitor rtiol d simplif. Emple : For > 0, simplif
11 Emple : Simplif: Multipl ot te umertor d deomitor te cojugte of te deomitor d simplif. ) ) ) ) Te et emple uses te sme procedure. Emple : Simplif: Assume ll vriles re positive ) ) ) ) ) )
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