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 ) ) ) ) ) )

Assessment Center Elementary Algebra Study Guide for the ACCUPLACER (CPT)

Assessment Center Elementary Algebra Study Guide for the ACCUPLACER (CPT) Assessmet Ceter Elemetr Alger Stud Guide for the ACCUPLACER (CPT) The followig smple questios re similr to the formt d cotet of questios o the Accuplcer Elemetr Alger test. Reviewig these smples will give