7.1 Finding Rational Solutions of Polynomial Equations


 Archibald Ryan
 3 years ago
 Views:
Transcription
1 4 Locker LESSON 7. Fidig Ratioal Solutios of Polyomial Equatios Name Class Date 7. Fidig Ratioal Solutios of Polyomial Equatios Essetial Questio: How do you fid the ratioal roots of a polyomial equatio? Resource Locker Commo Core Math Stadards The studet is expected to: AAPR. Kow ad apply the Remaider Theorem: For a polyomial p(x) ad a umber a, the remaider o divisio by x a is p(a), so p(a) = 0 if ad oly if (x a) is a factor of p(x). Also AAPR., ACED. Mathematical Practices MP. Reasoig Laguage Objective Explai to a parter how to idetify the factors of a polyomial fuctio. ENGAGE Essetial Questio: How do you fid the ratioal roots of a polyomial equatio? Use the Ratioal Root Theorem to idetify possible ratioal roots. Check each by usig sythetic substitutio. If a ratioal root is foud, repeat the process o the quotiet obtaied from the bottom row of the sythetic substitutio. Cotiue to fid ratioal roots i this way util the quotiet is quadratic, at which poit you ca try factorig to idetify the last two ratioal roots. Explore Relatig Zeros ad Coefficiets of Polyomial Fuctios The zeros of a polyomial fuctio ad the coefficiets of the fuctio are related. Cosider the polyomial fuctio ƒ (x) = (x + ) (x ) (x + ). A B C D E F Idetify the zeros of the polyomial fuctio. The zeros are x = , x =, ad x = . Multiply the factors to write the fuctio i stadard form. f (x) = (x + ) (x  ) (x + ) = ( x + x  x  ) (x + ) = ( x + x  ) (x + ) = x + x + x + x  x  6 = x + 4 x + x  6 How are the zeros of ƒ (x) related to the stadard form of the fuctio? Each of the zeros of the polyomial fuctio is a factor of the costat term i the stadard form. Now cosider the polyomial fuctio g (x) = (x + ) (4x  5) (6x  ). Idetify the zeros of this fuctio. The zeros are x =  _, ad x = _ 4 6., x = 5_ Multiply the factors to write the fuctio i stadard form. g (x) = (x + ) (4x  5) (6x  ) = (8 x  0x + x  5) (6x  ) = (8 x + x  5) (6x  ) = 48 x  8 x + x  x  90x + 5 = 48 x + 4 x  9x + 5 How are the zeros of g (x) related to the stadard form of the fuctio? Each of the umerators of the zeros is a factor of the costat term, 5, ad each of the deomiators is a factor of the leadig coefficiet, 48. PREVIEW: LESSON PERFORMANCE TASK View the Egage sectio olie. Discuss the photo ad how the umber of tourists i ay give year ca vary depedig o may factors. The preview the Lesso Performace Task. A_MNLESE85894_UM07L.idd 4 Module 7 4 Lesso DO NOT EDITChages must be made through File ifo CorrectioKey=NLA;CAA Name Class Date 7. Fidig Ratioal Solutios of Polyomial Equatios Essetial Questio: How do you fid the ratioal roots of a polyomial equatio? AAPR. Kow ad apply the Remaider Theorem: For a polyomial p (x) ad a umber a, the remaider o divisio by x  a is p (a), so p (a) = 0 if ad oly if (x  a) is a factor of p (x). Also AAPR., ACED. Explore Relatig Zeros ad Coefficiets of Polyomial Fuctios The zeros of a polyomial fuctio ad the coefficiets of the fuctio are related. Cosider the polyomial fuctio ƒ (x) = (x + ) (x )(x + ). Idetify the zeros of the polyomial fuctio. Multiply the factors to write the fuctio i stadard form. f (x) = (x + ) (x  ) (x + ) How are the zeros of ƒ (x) related to the stadard form of the fuctio? = (x + x  x  ) (x + ) = (x + x  ) (x + ) = x + x + x + x  x  6 Now cosider the polyomial fuctio g (x) = (x + ) (4x  5) (6x  ). Idetify the zeros of this fuctio. Multiply the factors to write the fuctio i stadard form. = x + 4 x + x  6 The zeros are x = , x =, ad x = . How are the zeros of g (x) related to the stadard form of the fuctio? Resource Each of the zeros of the polyomial fuctio is a factor of the costat term i the stadard form. The zeros are x =  _, x = 5_, ad x = _ g (x) = (x + ) (4x  5) (6x  ) 6. = (8 x  0x + x  5) (6x  ) = (8 x + x  5) (6x  ) = 48 x  8 x + x  x  90x + 5 = 48 x + 4 x  9x + 5 Each of the umerators of the zeros is a factor of the costat term, 5, ad each of the deomiators is a factor of the leadig coefficiet, 48. Module 7 4 Lesso A_MNLESE85894_UM07L.idd 4 /9/4 :9 PM HARDCOVER PAGES Tur to these pages to fid this lesso i the hardcover studet editio. /9/4 :7 PM 4 Lesso 7.
2 Reflect. I geeral, how are the zeros of a polyomial fuctio related to the fuctio writte i stadard form? Each of the umerators of the zeros is a factor of the costat term. Each of the deomiators of the zeros is a factor of the leadig coefficiet.. Discussio Does the relatioship from the first Reflect questio hold if the zeros are all itegers? Explai. Yes; If the zeros are all itegers, each of them ca be writte with a deomiator of. Each of the umerators is still a factor of the costat term.. If you use the zeros, you ca write the factored form of g (x) as g (x) = (x + )(x )(x  6 ), rather tha as g (x) = (x + ) (4x  5) (6x  ). What is the relatioship of the factors betwee the two forms? Give this relatioship i a geeral form. I each factor, the deomiator of the fractio becomes the coefficiet of the variable. I geeral, if the zero is  b_ a, the factor ca be writte as (ax + b). Explai Fidig Zeros Usig the Ratioal Zero Theorem If a polyomial fuctio p (x) is equal to ( a x + b ) ( a x + b ) ( a x + b ), where a, a, a, b, b, ad b are itegers, the leadig coefficiet of p (x) will be the product a a a ad the costat term will be b the product b b b. The zeros of p (x) will be the ratioal umbers  a,  b a,  b a. Comparig the zeros of p (x) to its coefficiet ad costat term shows that the umerators of the polyomial s zeros are factors of the costat term ad the deomiators of the zeros are factors of the leadig coefficiet. This result ca be geeralized as the Ratioal Zero Theorem. Ratioal Zero Theorem If p (x) is a polyomial fuctio with iteger coefficiets, ad if is a zero of p (x) (p( _ the m is a factor of the costat term of p (x) ad is a factor of the leadig coefficiet of p(x). ) = 0 ), Example Fid the ratioal zeros of the polyomial fuctio; the write the fuctio as a product of factors. Make sure to test the possible zeros to fid the actual zeros of the fuctio. ƒ (x) = x + x  9x  0 a. Use the Ratioal Zero Theorem to fid all possible ratioal zeros. Factors of 0: ±, ±, ±4, ±5, ±0, ±0 b. Test the possible zeros. Use a sythetic divisio table to orgaize the work. I this table, the first row (shaded) represets the coefficiets of the polyomial, the first colum represets the divisors, ad the last colum represets the remaiders Module 7 4 Lesso PROFESSIONAL DEVELOPMENT EXPLORE Relatig Zeros ad Coefficiets of Polyomial Fuctios INTEGRATE TECHNOLOGY Studets have the optio of completig the Explore activity either i the book or olie. QUESTIONING STRATEGIES What is the relatioship betwee the factors of a polyomial fuctio ad the zeros of the fuctio? The zeros are the values of x foud by settig each factor equal to 0 ad solvig for x. If a zero of a polyomial fuctio is 7, what do you kow about the coefficiets whe the polyomial is writte i stadard form? 7 is a factor of the costat term ad is a factor of the leadig coefficiet. EXPLAIN Fidig Zeros Usig the Ratioal Zero Theorem QUESTIONING STRATEGIES Is every zero of a polyomial fuctio represeted i the set of umbers give by the Ratioal Zero Theorem? No. The Ratioal Zero Theorem gives oly those zeros that are ratioal umbers. A polyomial fuctio ca also have zeros that are irratioal umbers or imagiary umbers. A_MNLESE85894_UM07L 4 Itegrate Mathematical Practices This lesso provides a opportuity to address Mathematical Practice MP., which calls for studets to traslate betwee multiple represetatios ad to reaso abstractly ad quatitatively. Studets explore the relatioship betwee the factors of a polyomial fuctio ad its zeros. They lear how to idetify the zeros give the factors, ad the factors give the zeros. They the explore the relatioships betwee the ratioal zeros of a fuctio ad its leadig coefficiet ad costat term, establishig the Ratioal Zero Theorem. 6/0/4 0:4 AM Fidig Ratioal Solutios of Polyomial Equatios 4
3 AVOID COMMON ERRORS Some studets may forget to iclude ad i their list of possible ratioal zeros. You may wat to suggest that they write these first so that they are ot iadvertetly left off the list. QUESTIONING STRATEGIES If the leadig coefficiet of a polyomial fuctio with iteger coefficiets is, what ca you coclude about the fuctio s ratioal zeros? Explai your reasoig. They must be itegers, because whe you apply the Ratioal Zero Theorem, ca equal oly or i. c. Factor the polyomial. The sythetic divisio by 4 results i a remaider of 0, so 4 is a zero ad the polyomial i factored form is give as follows: (x  4) ( x + 6x + 5) = 0 (x  4) (x + 5) (x + ) = 0 x = 4, x = 5, or x =  The zeros are x = 4, x = 5, ad x = . B ƒ (x) = x 44 x  7 x + x + 4 a. Use the Ratioal Zero Theorem to fid all possible ratioal zeros. Factors of 4: ±, ±, ±, ± 4, ± 6, ± 8, ±, ± 4 b. Test the possible zeros. Use a sythetic divisio table c. Factor the polyomial. The sythetic divisio by results i a remaider of 0, so is a zero ad the polyomial i factored form is give as follows: (x  )( x  x  0 x  8 ) = 0 d. Use the Ratioal Zero Theorem agai to fid all possible ratioal zeros of g (x) = x  x  0 x  8. Factors of 8: ±, ±, ± 4, ± e. Test the possible zeros. Use a sythetic divisio table f. Factor the polyomial. The sythetic divisio by 4 results i a remaider of 0, so 4 is a zero ad the polyomial i factored form is: (x  )(x  4 )( x + x + ) = 0 (x  )(x  4 )(x + )(x + ) = 0 x =, x = 4, x = , or x =  The zeros are x =, x = 4, x = , ad x = . Module 7 4 Lesso A_MNLESE85894_UM07L.idd 4 COLLABORATIVE LEARNING Small Group Activity Have studets work i groups of 4 studets. Istruct each group to create a fifthdegree polyomial fuctio with ratioal zeros, ot all of which are itegers. Ask them to write their fuctios i stadard from. Have groups exchage fuctios, ad have each group create a poster showig how to apply the Ratioal Zero Theorem to fid the zeros of the fuctio. Studets posters should also show verificatio that each umber is ideed a zero of the fuctio. 7/7/4 0:4 AM 4 Lesso 7.
4 Reflect 4. How is usig sythetic divisio o a 4 th degree polyomial to fid its zeros differet tha usig sythetic divisio o a rd degree polyomial to fid its zeros? To fid the zeros of a 4 th degree polyomial usig sythetic divisio, you eed to use sythetic divisio to reduce that polyomial to a rd degree polyomial ad the use sythetic divisio agai to reduce that polyomial to a quadratic polyomial that ca be factored, if possible. 5. Suppose you are tryig to fid the zeros the fuctio ƒ (x) = x +. Would it be possible to use sythetic divisio o this polyomial? Why or why ot? It would ot be possible to fid the zeros of this polyomial usig sythetic substitutio because the fuctio has o ratioal roots, oly complex roots. 6. Usig sythetic divisio, you fid that is a zero of ƒ (x) = x + x  x + 6. The quotiet from the sythetic divisio array for ƒ ( ) is x + x . Show how to write the factored form of ƒ (x) = x + x  x + 6 usig iteger coefficiets. Usig as a zero ad the quotiet x + x  you ca write f (x) = x + x  x + 6 as f (x) = ( x  _ ) ( x + x  ). f (x) = ( x  _ ) ( x + x  ) = ( x  _ ) () ( x + x  6) = (x  ) ( x + x  6) = (x  ) (x + ) (x  ) INTEGRATE MATHEMATICAL PRACTICES Focus o Math Coectios MP. Remid studets that a zero of a fuctio is a umber from the domai that the fuctio pairs with 0. Discuss that, for this reaso, a graph of the fuctio will have a xitercept at each zero. Studets ca the make a cocrete coectio betwee the ratioal zeros they idetify for a fuctio, ad the role the zeros play i the graph of the fuctio. Your Tur 7. Fid the zeros of ƒ (x) = x + x  x 5. a. Use the Ratioal Zero Theorem. Factors of 5: ±, ±, ±5, ±5 b. Test the possible zeros to fid oe that is actually a zero c. Factor the polyomial usig as a zero. (x  ) ( x + 6x + 5) = 0 (x  ) (x + ) (x + 5) = 0 x =, x = , or x = 5 The zeros are x =, x = , ad x = 5. Module 7 44 Lesso DIFFERENTIATE INSTRUCTION Visual Cues A_MNLESE85894_UM07L.idd 44 Ecourage studets to circle the leadig coefficiet i the fuctio ad to write is a factor of above it, ad to circle the costat term i the fuctio ad to write m is a factor of above it. This will be helpful whe applyig the Ratioal Zero Theorem, ad will keep studets from erroeously writig the reciprocals of the possible ratioal zeros, especially sice the usages of m ad appear i reverse alphabetical order with respect to the fuctio. 0/6/4 : PM Fidig Ratioal Solutios of Polyomial Equatios 44
5 EXPLAIN Solvig a RealWorld Problem Usig the Ratioal Root Theorem CONNECT VOCABULARY Explai how the words zeros ad roots (or solutios) have similar meaigs but are used i differet cotexts. The zeros of a fuctio are the roots (or solutios) of the related equatio. QUESTIONING STRATEGIES Why is it ecessary to rewrite the equatio so that it is equal to 0? I order to fid the roots of a equatio usig the Ratioal Root Theorem, the equatio must be i the form p (x) = 0. What iformatio is obtaied by applyig the Ratioal Zero Theorem to a polyomial fuctio? A list of all possible ratioal zeros of the fuctio Explai Solvig a RealWorld Problem Usig the Ratioal Root Theorem Sice a zero of a fuctio ƒ (x) is a value of x for which ƒ (x) = 0, fidig the zeros of a polyomial fuctio p (x) is the same thig as fid the solutios of the polyomial equatio p (x) = 0. Because a solutio of a polyomial equatio is kow as a root, the Ratioal Zero Theorem ca be also expressed as the Ratioal Root Theorem. Ratioal Root Theorem If the polyomial p (x) has iteger coefficiets, the every ratioal root of the polyomial equatio p (x) = 0 ca be writte i the form _, where m is a factor of the costat term of p (x) ad is a factor of the leadig coefficiet of p (x). Egieerig A pe compay is desigig a gift cotaier for their ew premium pe. The marketig departmet has desiged a pyramidal box with a rectagular base. The base width is ich shorter tha its base legth ad the height is iches taller tha times the base legth. The volume of the box must be 6 cubic iches. What are the dimesios of the box? Graph the volume fuctio ad the lie y = 6 o a graphig calculator to check your solutio. A. Aalyze Iformatio The importat iformatio is that the base width must be ich shorter tha the base legth, the height must be iches taller tha times the base legth, ad the box must have a volume of 6 cubic iches. B. Formulate a Pla Write a equatio to model the volume of the box. Let x represet the base legth i iches. The base width is height is x +, or (x + ). lw h = V ( x )(x  )()(x + ) = 6 x  x  6 = 0 x  ad the History i the markig Module 7 45 Lesso A_MNLESE85894_UM07L 45 6/8/4 : PM 45 Lesso 7.
6 C. Solve Use the Ratioal Root Theorem to fid all possible ratioal roots. Factors of 6: ±, ±, ±, ± 6 Test the possible roots. Use a sythetic divisio table Factor the polyomial. The sythetic divisio by results i a remaider of 0, so is a root ad the polyomial i factored form is as follows: ( x  ) ( x + x + ) = 0 The quadratic polyomial produces oly complex roots, so the oly possible aswer for the base legth is iches. The base width is ich ad the height is 9 iches. INTEGRATE MATHEMATICAL PRACTICES Focus o Critical Thikig MP. Prompt studets to recogize that ay ratioal roots foud by factorig the resultig quadratic polyomial must be umbers that were idetified as possible ratioal roots iitially. This may help them to catch errors i factorig, or i performig the sythetic divisio. D. Justify ad Evaluate The xcoordiates of the poits where the graphs of two fuctios, f ad g, itersect is the solutio of the equatio f (x) = g (x). Usig a graphig calculator to graph the volume fuctio ad y = 6 results i the graphs itersectig at the poit (, 6). Sice the xcoordiate is, the aswer is correct. Your Tur 8. Egieerig A box compay is desigig a ew rectagular gift cotaier. The marketig departmet has desiged a box with a width iches shorter tha its legth ad a height iches taller tha its legth. The volume of the box must be 56 cubic iches. What are the dimesios of the box? A. The box width must be iches shorter tha the legth, the height must be iches taller tha the width, ad the box must have a volume of 56 cubic iches. B. Let x represet the legth i iches. The width is x  ad the height is x +. lwh = V (x) (x  ) (x + ) = 56 x + x  6x = 56 x + x  6x  56 = 0 Module 7 46 Lesso A_MNLESE85894_UM07L 46 6/7/4 0: PM Fidig Ratioal Solutios of Polyomial Equatios 46
7 ELABORATE INTEGRATE MATHEMATICAL PRACTICES Focus o Techology MP.5 Have studets discuss how they could use a graphig utility to help determie which umbers from their list of possible ratioal zeros are more likely tha others to be zeros. Studets should recogize that they ca use the xitercepts of the graph to help them focus i o which umbers o their lists are good cadidates to test as possible zeros. QUESTIONING STRATEGIES If a cubic fuctio has oly oe ratioal root, what will be true about the quadratic polyomial quotiet that results from sythetic divisio by the ratioal root? It will ot be factorable over the set of itegers. SUMMARIZE THE LESSON How ca you use the Ratioal Root Theorem to fid the ratioal solutios of a polyomial equatio? You ca write the equatio i the form p (x) = 0, ad the use the theorem to idetify possible roots of the equatio. These roots will be of the form p q. You ca the test the possible roots usig sythetic substitutio. If you ca reduce the polyomial to a quadratic, you ca try factorig the quadratic to fid ay other ratioal roots. C. Use the Ratioal Root Theorem. Factors of 56: ±, ±, ±4, ±7, ±8, ±4, ±8, ±56 Test the possible roots to fid oe that is actually a root. Use a sythetic divisio table. Elaborate p_ q Factor the polyomial. usig 4 as a root. (x  4) ( x + 5x + 4) = 0 The quadratic polyomial produces oly complex roots. The oly possible aswer for the legth is 4 iches. The width is iches ad the height is 7 iches. D. Usig a graphig calculator, the graphs itersect at (4, 56), which validates the aswer. 9. For a polyomial fuctio with iteger coefficiets, how are the fuctio s coefficiets ad ratioal zeros related? The ratioal zeros of a polyomial fuctio with iteger coefficiets are i the form _, where m is a factor of the costat term ad is a factor of the leadig coefficiet. 0. Describe the process for fidig the ratioal zeros of a polyomial fuctio with iteger coefficiets. Usig the Ratioal Zero Theorem to fid all possible ratioal zeros, test the possible zeros to fid oe that is actually a zero by usig a sythetic divisio table to orgaize the work ad factor the polyomial.. How is the Ratioal Root Theorem useful whe solvig a realworld problem about the volume of a object whe the volume fuctio is a polyomial ad a specific value of the fuctio is give? The theorem is useful i this case because it allows you to fid the ratioal roots of the polyomial equatio created whe you set the volume fuctio equal to the give value. By rewritig the equatio so that oe side is 0, you ca use the Ratioal Root Theorem to fid the dimesio give by the variable ad the fid the other dimesios.. Essetial Questio CheckI What does the Ratioal Root Theorem fid? The Ratioal Root Theorem fids the possible ratioal roots of a polyomial equatio. Module 7 47 Lesso A_MNLESE85894_UM07L 47 LANGUAGE SUPPORT Commuicatig Math Have studets work i pairs. Istruct oe studet to write a polyomial fuctio i factor form. Have the secod studet idetify the zeros of the fuctio ad explai why they are the zeros. The studets switch roles ad repeat the process. Repeat the example from the lesso to provide a format. 6/7/4 0: PM 47 Lesso 7.
8 Evaluate: Homework ad Practice Fid the ratioal zeros of each polyomial fuctio. The write each fuctio i factored form. Olie Homework Hits ad Help Extra Practice EVALUATE. ƒ (x) = x x 0x 8. ƒ (x) = x + x  x  60 Factors of 8 : ±, ±, ±4, ±8 Factors of 60 : ±, ±, ±, ±4, ±5, ±6, 4 is a zero. ±0, ±, ±5, ±0, ±0, ±60 5 is a zero. (x  4) ( x + x + ) = 0 (x  4) (x + ) (x + ) = 0 x = 4, x = , or x =  f (x) = (x  4) (x + ) (x + ) (x  5) ( x + 7x + ) = 0 (x  5) (x + ) (x + 4) = 0 x = 5, x = , or x = 4 f (x) = (x  5) (x + ) (x + 4). j (x) = x  x  x g (x) = x  9 x + x 5 Factors of 6 : ±, ±, ±, ±6 is a zero. is a zero. (x  ) ( x + 5x + ) = 0 (x  ) (x + ) (x + ) = 0 x =, x =  _, or x =  j (x) = (x  ) (x + ) (x + ) Factors of 5 : ±, ±, ±5, ±5 (x  ) ( x  8x + 5) = 0 (x  ) (x 5) (x  ) = 0 x =, x = 5, or x = g (x) = (x  ) (x  5) (x  ) ASSIGNMENT GUIDE Cocepts ad Skills Explore Relatig Zeros ad Coefficiets of Polyomial Fuctios Example Fidig Zeros Usig the Ratioal Zero Theorem Example Solvig a RealWorld Problem Usig the Ratioal Root Theorem Practice Exercise 7 Exercises Exercises 6 5. h (x) = x  5 x + x h (x) = 6 x  7 x  9x Factors of 8 : ±, ±, ±4, ±8 is a zero. is a zero. (x  ) ( x  x  4) = 0 (x  ) (x  4) (x + ) = 0 x =, x = 4, or x =  h (x) = (x  ) (x  4) (x + ) Factors of : ±, ± (x  ) ( 6x + 5x + ) = 0 (x  ) (x + ) (x + ) = 0 x =, x =  _, or x =  _ h (x) = (x  ) (x + ) (x + ) 7. s (x) = x  x x + 8. t (x) = x + x 8x Factors of : ± Factors of : ±, ±, ±, ±4, ±6, ± is a zero. is a zero. (x  ) ( x  ) = 0 (x  ) (x + ) (x  ) = 0 x = or x =  s (x) = (x  ) (x + ) (x  ) (x  ) ( x + 4x + 4) = 0 (x  ) (x + ) (x + ) = 0 x = or x =  t (x) = (x  ) (x + ) (x + ) INTEGRATE MATHEMATICAL PRACTICES Focus o Patters MP.8 Studets ca use patters i the sigs of the terms i the polyomial fuctio to help them decide which of the possible ratioal zeros to test. For example, if the sigs of the terms i the polyomial fuctio (or i the quotiet after dividig sythetically) are all positive, studets eed ot check ay positive umbers o their lists. Module 7 48 Lesso A_MNLESE85894_UM07L 48 Exercise Depth of Kowledge (D.O.K.) Mathematical Practices 6/0/4 0:7 AM Recall of Iformatio MP.5 Usig Tools 7 Skills/Cocepts MP.4 Modelig 8 9 Skills/Cocepts MP. Logic 0 Strategic Thikig MP. Reasoig Strategic Thikig MP. Logic Fidig Ratioal Solutios of Polyomial Equatios 48
9 AVOID COMMON ERRORS Studets may icorrectly coclude that a polyomial fuctio that has ratioal zeros has oly real zeros. Explai that the fuctio may have irratioal zeros as well, ad irratioal zeros are real zeros. CONNECT VOCABULARY Have studets, i their ow words, explai how the Ratioal Zero Theorem ad the Ratioal Root Theorem are related (for example, a solutio of a polyomial equatio is ofte called a root). 9. k (x) = x x  x 7x + 0. g (x) = x 46 x + x  6x Factors of : ±, ±, ±, ±4, ±6, ± is a zero. Factor the polyomial. (x ) ( x + 6 x + 5x ) Factors of : ±, ±, ±, ±4, ±6, ± is a zero. (x  ) (x  ) ( x + 7x + ) = 0 (x  ) (x  ) (x + ) (x + 4) = 0 x =, x = , or x = 4 k (x) = (x  ) (x  ) (x + ) (x + 4) g (x) = x ( x  6 x + x  6) Factors of 6 : ±, ±, ±, ±6 is a zero. (x) (x  ) ( x  5x + 6) = 0 (x) (x  ) (x  ) (x  ) = 0 x =, x = 0, x =, or x = g (x) = (x) (x  ) (x  ) (x  ). h (x) = x 4  x  x + 4x + 4. ƒ (x) = x 45 x + 4 Factors of 4 : ±, ±, ±4 is a zero. is a zero. (x ) ( x x ) Factors of  : ±, ± is a zero. is a zero. Factors of 4 : ±, ±, ±4 f (x) = (x  ) ( x + x 4x 4) Factors of 4 : ±, ±, ±4 (x  ) (x  ) ( x + x + ) = 0 (x  ) (x  ) (x + ) (x + ) = 0 x =  or x = h (x) = (x  ) (x  ) (x + ) (x + ) (x  ) (x  ) ( x + x + ) = 0 (x  ) (x  ) (x + ) (x + ) = 0 x =, x =, x = , or x =  f (x) = (x  ) (x  ) (x + ) (x + ). Maufacturig A laboratory supply compay is desigig a ew rectagular box i which to ship glass pipes. The compay has created a box with a width iches shorter tha its legth ad a height 9 iches taller tha twice its legth. The volume of each box must be 45 cubic iches. What are the dimesios? Let x represet the legth i iches. The the width is x  ad the height is x + 9. lwh = V (x) (x  ) (x + 9) = 0 x + 5 x  8x = 45 x + 5 x  8x  45 = 0 Factors of 45: ±, ±, ±5, ±9, ±5, ±45 is a root. (x  ) ( x + x + 5) = 0, so (x  ) (x + 5) (x + ) = 0 x =, x = 5, or x =  Legth caot be egative. The legth must be is iches. The width is ich ad the height is 5 iches. Module 7 49 Lesso A_MNLESE85894_UM07L 49 6/0/4 0:46 AM 49 Lesso 7.
10 4. Egieerig A atural history museum is buildig a pyramidal glass structure for its tree sake exhibit. Its research team has desiged a pyramid with a square base ad with a height that is yards more tha a side of its base. The volume of the pyramid must be 47 cubic yards. What are the dimesios? Let x represet the side of the square base i yards. The height is x +. _ lwh = V _ (x) (x)(x + ) = 47 _ ( x + x ) = 47 x + x = 44 x + x  44 = 0 Factors of 44: ±, ±, ±7, ±9, ±, ±49, ±6, ±47, ±44 7 is a root. (x  7) ( x + 9x + 6) = 0 The quadratic factor produces oly complex roots. So, each side of the base is 7 yards ad the height is 9 yards. CRITICAL THINKING Discuss with studets why the Ratioal Root Theorem works, by applyig it to a quadratic equatio, such as x + x  5 = 0, ad showig how the process of solvig the equatio by factorig focuses o the factors of p ad q i a way that is similar to the process of the Ratioal Root Theorem. Focus studets attetio o how p is the product of the first coefficiets of the factors, ad q is the product of the costat terms of the factors. 5. Egieerig A paper compay is desigig a ew, pyramidshaped paperweight. Its developmet team has decided that to make the legth of the paperweight 4 iches less tha the height ad the width of the paperweight iches less tha the height. The paperweight must have a volume of cubic iches. What are the dimesios of the paperweight? Let x represet the height i iches. The legth is x  4 ad the width is x . _ lwh = V _ (x  4) (x  ) (x) = _ ( x  7x + x) = x  7 x + x = 6 x  7 x + x  6 = 0 Factors of 6: ±, ±, ±, ±4, ±6, ±9, ±, ±8, ±6 6 is a root. (x  6) ( x  x + 6) = 0 The quadratic factor produces oly complex roots. So, the height is 6 iches, the legth is iches, ad the width is iches. Image Credits: James Kigma/Shutterstock Module 7 50 Lesso A_MNLESE85894_UM07L.idd 50 /9/4 :7 PM Fidig Ratioal Solutios of Polyomial Equatios 50
11 PEERTOPEER DISCUSSION Ask studets to discuss with a parter how the Ratioal Root Theorem, i cojuctio with the Zero Product Property, eables them to solve realworld problems that ca be modeled by polyomial equatios. The Ratioal Root Theorem ca be used to idetify possible solutios. Idetifyig oe or more of the solutios from the list of possible solutios ca help you to write the equatio i factored form. You ca the use the Zero Product Property to set each factor equal to zero ad solve for other possible solutios. 6. Match each set of roots with its polyomial fuctio. B A. x =, x =, x = 4 ƒ (x) = (x + ) (x + 4) ( x _ B. x =, x = 4, x = _ C. x = _, x = _ 5 4, x = _ 7 D. x = 4_ 5, x = _ 6 C A ƒ (x) = ( x ) _ ( x _ 5 ) 4) ( x + _ 7 ) ƒ (x) = (x ) (x ) (x 4) 7, x = 4 D ƒ (x) = ( x + 4_ 5) ( x _ 7) 6 (x 4) 7. Idetify the zeros of ƒ (x) = (x + ) (x  4) (x  ), write the fuctio i stadard form, ad state how the zeros are related to the stadard form. The zeros of f (x) are x = , x = 4, ad x =. f (x) = (x + ) (x  4) (x  ) = ( x + x  4x  ) (x  ) = ( x  x  ) (x  ) = x  x  x + x  x + 6 = x  4 x  9x + 6 The zeros of f (x) are all factors of the costat term i the polyomial fuctio. JOURNAL Have studets describe how they could use the Ratioal Zero Theorem to write a polyomial fuctio i itercept form. H.O.T. Focus o Higher Order Thikig 8. Critical Thikig Cosider the polyomial fuctio g (x) = x  6 x + πx + 5. Is it possible to use the Ratioal Zero Theorem ad sythetic divisio to factor this polyomial? Explai. No; it is ot possible because the fuctio cotais a term, πx, whose coefficiet is irratioal ad, therefore, ot a iteger. 9. Explai the Error Sabria was told to fid the zeros of the polyomial fuctio h (x) = x (x  4) (x + ). She stated that the zeros of this polyomial are x = 0, x = 4, ad x =. Explai her error. For ay factor (ax + b), a zero occurs at  b_ a. Sabria forgot to iclude the egative sig whe covertig from her factors to the zeros. 0. Justify Reasoig If _ c is a ratioal zero of a polyomial fuctio p (x), explai why b bx  c must be a factor of the polyomial. Sice p ( c_ b) = 0, x  c_ is a factor of p (x) by the Factor Theorem. So, b p (x) = ( x  b) c_ q (x) ad p (x) = b_ b( x  b) c_ q (x) = _ (bx  c) q (x), which b shows that bx  c is a factor of p (x).. Justify Reasoig A polyomial fuctio p (x) has degree, ad its zeros are, 4, ad 6. What do you thik is the equatio of p (x)? Do you thik there could be more tha oe possibility? Explai. p (x) = (x + ) (x  4) (x  6) ; ay costat multiple of p (x) will also have degree ad the same zeros, so the equatio ca be ay fuctio of the form p (x) = a (x + ) (x  4) (x  6) where a 0. Module 7 5 Lesso A_MNLESE85894_UM07L 5 6/0/4 0:5 AM 5 Lesso 7.
12 Lesso Performace Task For the years from 00 00, the umber of Americas travelig to other coutries by plae ca be represeted by the polyomial fuctio A (t) = 0 t 448 t t  40t +,600, where A is the umber of thousads of Americas travelig abroad by airplae ad t is the umber of years sice 00. I which year were there 40,000,000 Americas travelig abroad? Use the Ratioal Root Theorem to fid your aswer. [Hit: cosider the fuctio s domai ad rage before fidig all possible ratioal roots.] A (t) = 0 t 448 t t  40t +,600 40,000 = 0 t 448 t t  40t +,600 0 = 0 t 448 t t  40t Factors of betwee 0 ad 9:,, 4, 5, 8. Test the possible roots: (x  5) (0 x  8 x + 0x + 80) Factors of 80 betwee 0 ad 9:,, 4, 5, 8. Test the possible roots: Because the cubic polyomial factor has o ratioal roots betwee 0 ad 9, x = 5 years returs the oly solutio. I other words, there were 40,000,000 Americas travelig overseas by air i 006. Image Credits: Paul Seheult/Eye Ubiquitous/Corbis AVOID COMMON ERRORS Some studets may set A (t) equal to 40,000,000, which is the umber give i the problem. Ask studets to check the uits of A. thousads of Americas. Have studets divide 40,000,000 by,000 to get the correct value for A, 40,000. More precisely, A is 40,000 thousads of Americas. QUESTIONING STRATEGIES Why is it useful to kow a fuctio s domai whe solvig for the roots? If the domai cosists oly of ratioal umbers, the the roots must be ratioal. For example, if the domai cosists of the itegers from 0 to 9, the the roots must be ratioal because itegers are ratioal umbers. Why does the domai cosist oly of itegers? The domai is the umber of years sice 00. The fuctio oly makes sese for iteger values. Module 7 5 Lesso EXTENSION ACTIVITY A_MNLESE85894_UM07L 5 Have studets research the factors that affect tourist umbers, such as chages i ecoomic status, or the safety of a destiatio. Have studets discuss who might use a model of tourist umbers like A (t) ad how it might be used. Ask studets to describe situatios i which it would be useful to iput a value of t to calculate the umber of tourists, ad i what situatios it would be useful to do the iverse use a give umber of tourists ad solve for the roots. 6/9/5 :5 AM Scorig Rubric poits: Studet correctly solves the problem ad explais his/her reasoig. poit: Studet shows good uderstadig of the problem but does ot fully solve or explai his/her reasoig. 0 poits: Studet does ot demostrate uderstadig of the problem. Fidig Ratioal Solutios of Polyomial Equatios 5
23 The Remainder and Factor Theorems
 The Remaider ad Factor Theorems Factor each polyomial completely usig the give factor ad log divisio 1 x + x x 60; x + So, x + x x 60 = (x + )(x x 15) Factorig the quadratic expressio yields x + x x
More informationSoving Recurrence Relations
Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree
More informationTrigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is
0_0605.qxd /5/05 0:45 AM Page 470 470 Chapter 6 Additioal Topics i Trigoometry 6.5 Trigoometric Form of a Complex Number What you should lear Plot complex umbers i the complex plae ad fid absolute values
More informationFOUNDATIONS OF MATHEMATICS AND PRECALCULUS GRADE 10
FOUNDATIONS OF MATHEMATICS AND PRECALCULUS GRADE 10 [C] Commuicatio Measuremet A1. Solve problems that ivolve liear measuremet, usig: SI ad imperial uits of measure estimatio strategies measuremet strategies.
More informationExample 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).
BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook  Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly
More informationSequences and Series
CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their
More informationRepeating Decimals are decimal numbers that have number(s) after the decimal point that repeat in a pattern.
5.5 Fractios ad Decimals Steps for Chagig a Fractio to a Decimal. Simplify the fractio, if possible. 2. Divide the umerator by the deomiator. d d Repeatig Decimals Repeatig Decimals are decimal umbers
More informationTHE ARITHMETIC OF INTEGERS.  multiplication, exponentiation, division, addition, and subtraction
THE ARITHMETIC OF INTEGERS  multiplicatio, expoetiatio, divisio, additio, ad subtractio What to do ad what ot to do. THE INTEGERS Recall that a iteger is oe of the whole umbers, which may be either positive,
More informationSECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
More informationHere are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
This documet was writte ad copyrighted by Paul Dawkis. Use of this documet ad its olie versio is govered by the Terms ad Coditios of Use located at http://tutorial.math.lamar.edu/terms.asp. The olie versio
More informationMATH 083 Final Exam Review
MATH 08 Fial Eam Review Completig the problems i this review will greatly prepare you for the fial eam Calculator use is ot required, but you are permitted to use a calculator durig the fial eam period
More information3. Greatest Common Divisor  Least Common Multiple
3 Greatest Commo Divisor  Least Commo Multiple Defiitio 31: The greatest commo divisor of two atural umbers a ad b is the largest atural umber c which divides both a ad b We deote the greatest commo gcd
More informationMultiple Representations for Pattern Exploration with the Graphing Calculator and Manipulatives
Douglas A. Lapp Multiple Represetatios for Patter Exploratio with the Graphig Calculator ad Maipulatives To teach mathematics as a coected system of cocepts, we must have a shift i emphasis from a curriculum
More informationG r a d e. 2 M a t h e M a t i c s. statistics and Probability
G r a d e 2 M a t h e M a t i c s statistics ad Probability Grade 2: Statistics (Data Aalysis) (2.SP.1, 2.SP.2) edurig uderstadigs: data ca be collected ad orgaized i a variety of ways. data ca be used
More informationAP Calculus BC 2003 Scoring Guidelines Form B
AP Calculus BC Scorig Guidelies Form B The materials icluded i these files are iteded for use by AP teachers for course ad exam preparatio; permissio for ay other use must be sought from the Advaced Placemet
More information1. MATHEMATICAL INDUCTION
1. MATHEMATICAL INDUCTION EXAMPLE 1: Prove that for ay iteger 1. Proof: 1 + 2 + 3 +... + ( + 1 2 (1.1 STEP 1: For 1 (1.1 is true, sice 1 1(1 + 1. 2 STEP 2: Suppose (1.1 is true for some k 1, that is 1
More informationLesson 17 Pearson s Correlation Coefficient
Outlie Measures of Relatioships Pearso s Correlatio Coefficiet (r) types of data scatter plots measure of directio measure of stregth Computatio covariatio of X ad Y uique variatio i X ad Y measurig
More informationDetermining the sample size
Determiig the sample size Oe of the most commo questios ay statisticia gets asked is How large a sample size do I eed? Researchers are ofte surprised to fid out that the aswer depeds o a umber of factors
More informationMATHEMATICS P1 COMMON TEST JUNE 2014 NATIONAL SENIOR CERTIFICATE GRADE 12
Mathematics/P1 1 Jue 014 Commo Test MATHEMATICS P1 COMMON TEST JUNE 014 NATIONAL SENIOR CERTIFICATE GRADE 1 Marks: 15 Time: ½ hours N.B: This questio paper cosists of 7 pages ad 1 iformatio sheet. Please
More informationHypothesis testing. Null and alternative hypotheses
Hypothesis testig Aother importat use of samplig distributios is to test hypotheses about populatio parameters, e.g. mea, proportio, regressio coefficiets, etc. For example, it is possible to stipulate
More informationWHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER?
WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER? JÖRG JAHNEL 1. My Motivatio Some Sort of a Itroductio Last term I tought Topological Groups at the Göttige Georg August Uiversity. This
More informationListing terms of a finite sequence List all of the terms of each finite sequence. a) a n n 2 for 1 n 5 1 b) a n for 1 n 4 n 2
74 (4 ) Chapter 4 Sequeces ad Series 4. SEQUENCES I this sectio Defiitio Fidig a Formula for the th Term The word sequece is a familiar word. We may speak of a sequece of evets or say that somethig is
More informationI. Chisquared Distributions
1 M 358K Supplemet to Chapter 23: CHISQUARED DISTRIBUTIONS, TDISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad tdistributios, we first eed to look at aother family of distributios, the chisquared distributios.
More informationAP Calculus AB 2006 Scoring Guidelines Form B
AP Calculus AB 6 Scorig Guidelies Form B The College Board: Coectig Studets to College Success The College Board is a otforprofit membership associatio whose missio is to coect studets to college success
More informationCHAPTER 3 THE TIME VALUE OF MONEY
CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all
More information1 Correlation and Regression Analysis
1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio
More informationMathematical goals. Starting points. Materials required. Time needed
Level A1 of challege: C A1 Mathematical goals Startig poits Materials required Time eeded Iterpretig algebraic expressios To help learers to: traslate betwee words, symbols, tables, ad area represetatios
More informationIn nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008
I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces
More informationChapter 5: Inner Product Spaces
Chapter 5: Ier Product Spaces Chapter 5: Ier Product Spaces SECION A Itroductio to Ier Product Spaces By the ed of this sectio you will be able to uderstad what is meat by a ier product space give examples
More informationBINOMIAL EXPANSIONS 12.5. In this section. Some Examples. Obtaining the Coefficients
652 (1226) Chapter 12 Sequeces ad Series 12.5 BINOMIAL EXPANSIONS I this sectio Some Examples Otaiig the Coefficiets The Biomial Theorem I Chapter 5 you leared how to square a iomial. I this sectio you
More information1 Computing the Standard Deviation of Sample Means
Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.
More informationS. Tanny MAT 344 Spring 1999. be the minimum number of moves required.
S. Tay MAT 344 Sprig 999 Recurrece Relatios Tower of Haoi Let T be the miimum umber of moves required. T 0 = 0, T = 7 Iitial Coditios * T = T + $ T is a sequece (f. o itegers). Solve for T? * is a recurrece,
More information5.4 Amortization. Question 1: How do you find the present value of an annuity? Question 2: How is a loan amortized?
5.4 Amortizatio Questio 1: How do you fid the preset value of a auity? Questio 2: How is a loa amortized? Questio 3: How do you make a amortizatio table? Oe of the most commo fiacial istrumets a perso
More informationFactoring x n 1: cyclotomic and Aurifeuillian polynomials Paul Garrett <garrett@math.umn.edu>
(March 16, 004) Factorig x 1: cyclotomic ad Aurifeuillia polyomials Paul Garrett Polyomials of the form x 1, x 3 1, x 4 1 have at least oe systematic factorizatio x 1 = (x 1)(x 1
More informationNATIONAL SENIOR CERTIFICATE GRADE 11
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P NOVEMBER 007 MARKS: 50 TIME: 3 hours This questio paper cosists of 9 pages, diagram sheet ad a page formula sheet. Please tur over Mathematics/P DoE/November
More informationMath 114 Intermediate Algebra Integral Exponents & Fractional Exponents (10 )
Math 4 Math 4 Itermediate Algebra Itegral Epoets & Fractioal Epoets (0 ) Epoetial Fuctios Epoetial Fuctios ad Graphs I. Epoetial Fuctios The fuctio f ( ) a, where is a real umber, a 0, ad a, is called
More informationChapter 7: Confidence Interval and Sample Size
Chapter 7: Cofidece Iterval ad Sample Size Learig Objectives Upo successful completio of Chapter 7, you will be able to: Fid the cofidece iterval for the mea, proportio, ad variace. Determie the miimum
More informationMath C067 Sampling Distributions
Math C067 Samplig Distributios Sample Mea ad Sample Proportio Richard Beigel Some time betwee April 16, 2007 ad April 16, 2007 Examples of Samplig A pollster may try to estimate the proportio of voters
More informationBuilding Blocks Problem Related to Harmonic Series
TMME, vol3, o, p.76 Buildig Blocks Problem Related to Harmoic Series Yutaka Nishiyama Osaka Uiversity of Ecoomics, Japa Abstract: I this discussio I give a eplaatio of the divergece ad covergece of ifiite
More informationHypergeometric Distributions
7.4 Hypergeometric Distributios Whe choosig the startig lieup for a game, a coach obviously has to choose a differet player for each positio. Similarly, whe a uio elects delegates for a covetio or you
More informationGCE Further Mathematics (6360) Further Pure Unit 2 (MFP2) Textbook. Version: 1.4
GCE Further Mathematics (660) Further Pure Uit (MFP) Tetbook Versio: 4 MFP Tetbook Alevel Further Mathematics 660 Further Pure : Cotets Chapter : Comple umbers 4 Itroductio 5 The geeral comple umber 5
More informationBasic Elements of Arithmetic Sequences and Series
MA40S PRECALCULUS UNIT G GEOMETRIC SEQUENCES CLASS NOTES (COMPLETED NO NEED TO COPY NOTES FROM OVERHEAD) Basic Elemets of Arithmetic Sequeces ad Series Objective: To establish basic elemets of arithmetic
More informationCS103X: Discrete Structures Homework 4 Solutions
CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible sixfigure salaries i whole dollar amouts are there that cotai at least
More informationSEQUENCES AND SERIES CHAPTER
CHAPTER SEQUENCES AND SERIES Whe the Grat family purchased a computer for $,200 o a istallmet pla, they agreed to pay $00 each moth util the cost of the computer plus iterest had bee paid The iterest each
More informationSolving equations. Pretest. Warmup
Solvig equatios 8 Pretest Warmup We ca thik of a algebraic equatio as beig like a set of scales. The two sides of the equatio are equal, so the scales are balaced. If we add somethig to oe side of the
More informationTHE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n
We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample
More information3. If x and y are real numbers, what is the simplified radical form
lgebra II Practice Test Objective:.a. Which is equivalet to 98 94 4 49?. Which epressio is aother way to write 5 4? 5 5 4 4 4 5 4 5. If ad y are real umbers, what is the simplified radical form of 5 y
More informationSAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx
SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval
More informationEngineering 323 Beautiful Homework Set 3 1 of 7 Kuszmar Problem 2.51
Egieerig 33 eautiful Homewor et 3 of 7 Kuszmar roblem.5.5 large departmet store sells sport shirts i three sizes small, medium, ad large, three patters plaid, prit, ad stripe, ad two sleeve legths log
More informationYour organization has a Class B IP address of 166.144.0.0 Before you implement subnetting, the Network ID and Host ID are divided as follows:
Subettig Subettig is used to subdivide a sigle class of etwork i to multiple smaller etworks. Example: Your orgaizatio has a Class B IP address of 166.144.0.0 Before you implemet subettig, the Network
More informationSEQUENCES AND SERIES
Chapter 9 SEQUENCES AND SERIES Natural umbers are the product of huma spirit. DEDEKIND 9.1 Itroductio I mathematics, the word, sequece is used i much the same way as it is i ordiary Eglish. Whe we say
More informationMeasures of Spread and Boxplots Discrete Math, Section 9.4
Measures of Spread ad Boxplots Discrete Math, Sectio 9.4 We start with a example: Example 1: Comparig Mea ad Media Compute the mea ad media of each data set: S 1 = {4, 6, 8, 10, 1, 14, 16} S = {4, 7, 9,
More informationLaws of Exponents Learning Strategies
Laws of Epoets Learig Strategies What should studets be able to do withi this iteractive? Studets should be able to uderstad ad use of the laws of epoets. Studets should be able to simplify epressios that
More informationOverview. Learning Objectives. Point Estimate. Estimation. Estimating the Value of a Parameter Using Confidence Intervals
Overview Estimatig the Value of a Parameter Usig Cofidece Itervals We apply the results about the sample mea the problem of estimatio Estimatio is the process of usig sample data estimate the value of
More informationCREATIVE MARKETING PROJECT 2016
CREATIVE MARKETING PROJECT 2016 The Creative Marketig Project is a chapter project that develops i chapter members a aalytical ad creative approach to the marketig process, actively egages chapter members
More informationMaximum Likelihood Estimators.
Lecture 2 Maximum Likelihood Estimators. Matlab example. As a motivatio, let us look at oe Matlab example. Let us geerate a radom sample of size 00 from beta distributio Beta(5, 2). We will lear the defiitio
More informationMaking training work for your business
Makig traiig work for your busiess Itegratig core skills of laguage, literacy ad umeracy ito geeral workplace traiig makes sese. The iformatio i this pamphlet will help you pla for ad build a successful
More informationG r a d e. 5 M a t h e M a t i c s. Number
G r a d e 5 M a t h e M a t i c s Number Grade 5: Number (5.N.1) edurig uderstadigs: the positio of a digit i a umber determies its value. each place value positio is 10 times greater tha the place value
More informationCURIOUS MATHEMATICS FOR FUN AND JOY
WHOPPING COOL MATH! CURIOUS MATHEMATICS FOR FUN AND JOY APRIL 1 PROMOTIONAL CORNER: Have you a evet, a workshop, a website, some materials you would like to share with the world? Let me kow! If the work
More informationSection 11.3: The Integral Test
Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult
More information5: Introduction to Estimation
5: Itroductio to Estimatio Cotets Acroyms ad symbols... 1 Statistical iferece... Estimatig µ with cofidece... 3 Samplig distributio of the mea... 3 Cofidece Iterval for μ whe σ is kow before had... 4 Sample
More informationCHAPTER 7: Central Limit Theorem: CLT for Averages (Means)
CHAPTER 7: Cetral Limit Theorem: CLT for Averages (Meas) X = the umber obtaied whe rollig oe six sided die oce. If we roll a six sided die oce, the mea of the probability distributio is X P(X = x) Simulatio:
More informationBaan Service Master Data Management
Baa Service Master Data Maagemet Module Procedure UP069A US Documetiformatio Documet Documet code : UP069A US Documet group : User Documetatio Documet title : Master Data Maagemet Applicatio/Package :
More informationChapter 7 Methods of Finding Estimators
Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of
More informationNATIONAL SENIOR CERTIFICATE GRADE 12
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 04 MARKS: 50 TIME: 3 hours This questio paper cosists of 8 pages ad iformatio sheet. Please tur over Mathematics/P DBE/04 NSC Grade Eemplar INSTRUCTIONS
More informationElementary Theory of Russian Roulette
Elemetary Theory of Russia Roulette iterestig patters of fractios Satoshi Hashiba Daisuke Miematsu Ryohei Miyadera Itroductio. Today we are goig to study mathematical theory of Russia roulette. If some
More informationCS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations
CS3A Hadout 3 Witer 00 February, 00 Solvig Recurrece Relatios Itroductio A wide variety of recurrece problems occur i models. Some of these recurrece relatios ca be solved usig iteratio or some other ad
More information.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth
Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,
More informationConfidence Intervals. CI for a population mean (σ is known and n > 30 or the variable is normally distributed in the.
Cofidece Itervals A cofidece iterval is a iterval whose purpose is to estimate a parameter (a umber that could, i theory, be calculated from the populatio, if measuremets were available for the whole populatio).
More informationCase Study. Normal and t Distributions. Density Plot. Normal Distributions
Case Study Normal ad t Distributios Bret Halo ad Bret Larget Departmet of Statistics Uiversity of Wiscosi Madiso October 11 13, 2011 Case Study Body temperature varies withi idividuals over time (it ca
More informationChapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions
Chapter 5 Uit Aual Amout ad Gradiet Fuctios IET 350 Egieerig Ecoomics Learig Objectives Chapter 5 Upo completio of this chapter you should uderstad: Calculatig future values from aual amouts. Calculatig
More informationINFINITE SERIES KEITH CONRAD
INFINITE SERIES KEITH CONRAD. Itroductio The two basic cocepts of calculus, differetiatio ad itegratio, are defied i terms of limits (Newto quotiets ad Riema sums). I additio to these is a third fudametal
More information*The most important feature of MRP as compared with ordinary inventory control analysis is its time phasing feature.
Itegrated Productio ad Ivetory Cotrol System MRP ad MRP II Framework of Maufacturig System Ivetory cotrol, productio schedulig, capacity plaig ad fiacial ad busiess decisios i a productio system are iterrelated.
More informationThe analysis of the Cournot oligopoly model considering the subjective motive in the strategy selection
The aalysis of the Courot oligopoly model cosiderig the subjective motive i the strategy selectio Shigehito Furuyama Teruhisa Nakai Departmet of Systems Maagemet Egieerig Faculty of Egieerig Kasai Uiversity
More informationWeek 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable
Week 3 Coditioal probabilities, Bayes formula, WEEK 3 page 1 Expected value of a radom variable We recall our discussio of 5 card poker hads. Example 13 : a) What is the probability of evet A that a 5
More informationDepartment of Computer Science, University of Otago
Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS200609 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly
More informationChapter 6: Variance, the law of large numbers and the MonteCarlo method
Chapter 6: Variace, the law of large umbers ad the MoteCarlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value
More informationhp calculators HP 12C Statistics  average and standard deviation Average and standard deviation concepts HP12C average and standard deviation
HP 1C Statistics  average ad stadard deviatio Average ad stadard deviatio cocepts HP1C average ad stadard deviatio Practice calculatig averages ad stadard deviatios with oe or two variables HP 1C Statistics
More informationEngineering Data Management
BaaERP 5.0c Maufacturig Egieerig Data Maagemet Module Procedure UP128A US Documetiformatio Documet Documet code : UP128A US Documet group : User Documetatio Documet title : Egieerig Data Maagemet Applicatio/Package
More informationGrade 7 Mathematics. Support Document for Teachers
Grade 7 Mathematics Support Documet for Teachers G r a d e 7 M a t h e m a t i c s Support Documet for Teachers 2012 Maitoba Educatio Maitoba Educatio Cataloguig i Publicatio Data Grade 7 mathematics
More informationSolving Logarithms and Exponential Equations
Solvig Logarithms ad Epoetial Equatios Logarithmic Equatios There are two major ideas required whe solvig Logarithmic Equatios. The first is the Defiitio of a Logarithm. You may recall from a earlier topic:
More informationwhere: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return
EVALUATING ALTERNATIVE CAPITAL INVESTMENT PROGRAMS By Ke D. Duft, Extesio Ecoomist I the March 98 issue of this publicatio we reviewed the procedure by which a capital ivestmet project was assessed. The
More informationConfidence Intervals for One Mean
Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a
More informationNow here is the important step
LINEST i Excel The Excel spreadsheet fuctio "liest" is a complete liear least squares curve fittig routie that produces ucertaity estimates for the fit values. There are two ways to access the "liest"
More informationFast Fourier Transform
18.310 lecture otes November 18, 2013 Fast Fourier Trasform Lecturer: Michel Goemas I these otes we defie the Discrete Fourier Trasform, ad give a method for computig it fast: the Fast Fourier Trasform.
More informationTheorems About Power Series
Physics 6A Witer 20 Theorems About Power Series Cosider a power series, f(x) = a x, () where the a are real coefficiets ad x is a real variable. There exists a real oegative umber R, called the radius
More informationSimple Annuities Present Value.
Simple Auities Preset Value. OBJECTIVES (i) To uderstad the uderlyig priciple of a preset value auity. (ii) To use a CASIO CFX9850GB PLUS to efficietly compute values associated with preset value auities.
More informationApproximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find
1.8 Approximatig Area uder a curve with rectagles 1.6 To fid the area uder a curve we approximate the area usig rectagles ad the use limits to fid 1.4 the area. Example 1 Suppose we wat to estimate 1.
More informationUniversity of California, Los Angeles Department of Statistics. Distributions related to the normal distribution
Uiversity of Califoria, Los Ageles Departmet of Statistics Statistics 100B Istructor: Nicolas Christou Three importat distributios: Distributios related to the ormal distributio Chisquare (χ ) distributio.
More informationCME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 8
CME 30: NUMERICAL LINEAR ALGEBRA FALL 005/06 LECTURE 8 GENE H GOLUB 1 Positive Defiite Matrices A matrix A is positive defiite if x Ax > 0 for all ozero x A positive defiite matrix has real ad positive
More informationBENEFITCOST ANALYSIS Financial and Economic Appraisal using Spreadsheets
BENEITCST ANALYSIS iacial ad Ecoomic Appraisal usig Spreadsheets Ch. 2: Ivestmet Appraisal  Priciples Harry Campbell & Richard Brow School of Ecoomics The Uiversity of Queeslad Review of basic cocepts
More informationProperties of MLE: consistency, asymptotic normality. Fisher information.
Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout
More information1. C. The formula for the confidence interval for a population mean is: x t, which was
s 1. C. The formula for the cofidece iterval for a populatio mea is: x t, which was based o the sample Mea. So, x is guarateed to be i the iterval you form.. D. Use the rule : pvalue
More informationHow to read A Mutual Fund shareholder report
Ivestor BulletI How to read A Mutual Fud shareholder report The SEC s Office of Ivestor Educatio ad Advocacy is issuig this Ivestor Bulleti to educate idividual ivestors about mutual fud shareholder reports.
More informationFIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. 1. Powers of a matrix
FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. Powers of a matrix We begi with a propositio which illustrates the usefuless of the diagoalizatio. Recall that a square matrix A is diogaalizable if
More informationProfessional Networking
Professioal Networkig 1. Lear from people who ve bee where you are. Oe of your best resources for etworkig is alumi from your school. They ve take the classes you have take, they have bee o the job market
More informationBiology 171L Environment and Ecology Lab Lab 2: Descriptive Statistics, Presenting Data and Graphing Relationships
Biology 171L Eviromet ad Ecology Lab Lab : Descriptive Statistics, Presetig Data ad Graphig Relatioships Itroductio Log lists of data are ofte ot very useful for idetifyig geeral treds i the data or the
More informationEscola Federal de Engenharia de Itajubá
Escola Federal de Egeharia de Itajubá Departameto de Egeharia Mecâica PósGraduação em Egeharia Mecâica MPF04 ANÁLISE DE SINAIS E AQUISÇÃO DE DADOS SINAIS E SISTEMAS Trabalho 02 (MATLAB) Prof. Dr. José
More informationSystems Design Project: Indoor Location of Wireless Devices
Systems Desig Project: Idoor Locatio of Wireless Devices Prepared By: Bria Murphy Seior Systems Sciece ad Egieerig Washigto Uiversity i St. Louis Phoe: (805) 6985295 Email: bcm1@cec.wustl.edu Supervised
More information