Here are the coefficients of the terms listed above: 3,5,2,1,1 respectively.

Transcription

1 *. Operatios with Poloials: Let s start b defiig soe words. Ter: A ter is a uber, variable or the product of a uber ad variable(s). For eaple:,, z, a Coefficiet: A coefficiet is the ueric factor of the ter. Here are the coefficiets of the ters listed above:,,,, respectivel. A ooial is a poloial that cosists of eactl oe ter. For eaple,. A bioial is a poloial that cosists of eactl two ters. For eaple, +. A trioial is a poloial that cosists of eactl three ters. For eaple, 9 B defiitio, all ooials, bioials, ad trioials are also poloials. * Addig Poloials: To add poloials, reove the paretheses (or a groupig sbol), the cobie like ters. Like ters: are ters that cotai the sae variables raised to eactl the sae powers. Eaple : Add the followig poloials. 0 a. Solutio: 0 = 0 = 9 b. * Subtractig Poloials: To subtract poloials, reove the paretheses (or a groupig sbol) b chagig the sigs of the ters of the poloial beig subtracted, ad the cobie like ters. Eaple : Subtract the followig poloials. a. 0 Solutio: Whe subtractig poloials, the first thig we ll do is distribute the ius sig through the parethesis. This eas that we will chage the sig o ever ter i the secod poloial. Note that all we are reall doig here is ultiplig a - through the secod poloial usig the distributive law. After distributig the ius through the parethesis we agai cobie like ters.

2 So, 0 0 = 0 = 8 b. * Multiplig Poloials: To ultipl a two poloials, use the distributive propert ad ultipl each ter of oe poloial b each ter of the other poloial. The cobie like ters. ** Whe ultiplig Bioials, use FOIL (First-Outer-Ier-Last) Eaple : Multipl each of the followig. (a) (b) 0 Solutio: (a) This is a quick applicatio of the distributive propert. 8 (b) 0 Here we will use the FOIL ethod for ultiplig the two bioials *Special products: 0 a ba b a b a b a ab b a b a ab b a b a a b ab b a b a a b ab b Warig: Be careful ot to ake the followig istakes! a b a b a b a b a b a b a b a b These are ver coo istakes that studets ofte ake whe the first start learig how to ultipl poloials.

3 Eaple : Multipl each of the followig. Solutio: (a) (b) (c) (d) (e) (a) Use FOIL 9 9 I this case the iddle ters drop out. (b) 9 0 (c) 9 0 * Multiplig Three or More Poloials: To ultipl three or ore poloials, ore tha oe ethod a be eeded. 9 Eaple : Multipl Solutio: First Multipl the, ultipl * Dividig Poloials Usig Log Divisio (No Missig Ters): Eaple : Solutio:. Arrage the ters of both the divided ad the divisor i descedig order, ad check that there are o issig ters. This is alread doe for ou here.. Divide the first ter i the divided b the first ter i the divisor. The result is the first ter of the quotiet.. Multipl ever ter i the divisor b the first ter i the quotiet.

4 Write the resultig product beeath the divided with like ters lied up.. Subtract the product fro the divided.. Brig dow the et ter i the origial divided ad write it et to the reaider to for a ew divided.. Use this ew epressio as the divided ad repeat this process util the reaider ca o loger be divided. This will occur whe the degree of the reaider is less tha the degree of the divisor. Please see class otes for coplete solutio. *Dividig Poloials Usig Log Divisio (Missig Ters): Eaple 7: Divide b - Solutio: Let us first set up the proble. Recall that we eed to have the ters writte dow with the epoets i decreasig order ad to ake sure we do t ake a istakes we add i a issig ters with a zero coefficiet. Now we ask ourselves what we eed to ultipl to get the first ter i first poloial. I this case that is. So ultipl b ad subtract the results fro the first poloial

5 The ew poloial is called the reaider. We cotiue the process util the degree of the reaider is less tha the degree of the divisor, which is i this case. So, we eed to cotiue util the degree of the reaider is less tha. Recall that the degree of a poloial is the highest epoet i the poloial. Also, recall that a costat is thought of as a poloial of degree zero. Therefore, we ll eed to cotiue util we get a costat i this case. Here is the rest of the work of this eaple. Now, tr the followig probles:. Perfor the idicated operatios ad siplif whe eeded. (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) Please check HW probles as well.

6 . Factorig Poloials: Factorig is the reverse process of ultiplig. It is the process of writig a poloial as a product of factors. I other words, to factor a poloial eas to write it as a product of other poloials. * Methods for factorig Poloials: ) Greatest Coo Factor: The GCF for a poloial is the largest ooial that divides (is a factor of) each ter of the poloial. The first ethod for factorig poloials will be factorig out the greatest coo factor. Whe factorig i geeral this will also be the first ethod that we should tr as it will ofte siplif the proble. Eaple : Factor the followig: (a) 8 (b) (c) 8 0 See class otes (d) Solutio: (a) 8 The greatest coo factor of the ters 8 ad is. 8 (b) The greatest coo factor of the ters is. (c) 8 0 First we otice that we ca factor a out of ever ter. Also ote that we ca factor a out of ever ter. Here the is the factorig for this proble. 8 0 Note: We ca alwas check our factorig b ultiplig the ters out to ake sure we get the origial poloial. ) Factorig b Groupig: is used with poloials that have four or ore ters. Hit: ou will alost alwas use factorig b groupig o poloials with degree greater tha.

7 Eaple : Factor the followig: (a) 8 (b) Solutio (a) 8 I this case we group the first two ters ad the fial two ters as show here, ( ) ( 8) Now, factor a out of the first groupig ad a out of the secod groupig, which gives: 8 Notice that we ca factor out a coo factor of -, so the fial factored for. 8 Ad we are doe factorig b groupig. Note that if we ultipl our aswer out, we do get the origial poloial. (b) I this case we will do the sae iitial step, but this tie otice that both of the fial two ters are egative so we will factor out a - as well whe we group the. Doig this gives, ( ) ( ) Agai, we ca alwas distribute the - back through the parethesis to ake sure we get the origial poloial. At this poit we ca see that we ca factor a out of the first ter ad a out of the secod ter. This gives, = ( ) ( ) We ow have a coo factor that we ca factor out to coplete the proble. = ( )( ) Be careful. Whe the first ter of the secod group has a ius sig i frot of it, ou wat to put the ius i frot of the secod ( ). Whe ou do this ou eed to chage the sig of BOTH ters of the secod ( ) as show above. 7

8 * Special Factorizatio Patters: ) Differece of Squares: Usig the techique described above, factors as This tpe of epressio is called a "differece of squares.". Notice that, i this case, the "iddle ter" disappears whe ou ultipl these two factors. Special Note: I fact, a differece of squares will alwas factor accordig to the rule: a b a ba b Aother eaple is: 9p p p Warig: Whe workig with real ubers, a "su of squares" will ot factor accordig to this rule (or a other rule). Tr it o 9. Eaple: ) Differece ad Su of Cubes: We have just see that a differece of squares alwas factors. The sae is true for a "differece of cubes." For eaple: The geeral rule for a "differece of cubes" is: a a ba ab b b. Reeber, ou ca verif this rule b ultiplig out the factored for. Eaple: 8 7 Note: Although a su of squares will ot factor, a "su of cubes" will factor, for 7 9. eaple, 8

9 The geeral rule for a "su of cubes" is: a a ba ab b b. Eaple: 7 *Factorig Quadratic Poloials: First, ote that quadratic is aother ter for secod degree poloial. So we kow that the largest epoet i a quadratic poloial will be a. I these probles, we will be atteptig to factor quadratic poloials ito two first degree (hece forth liear) poloials. Util ou becoe good at these, we usuall ed up doig these b trial ad error although there are a couple of processes that ca ake the soewhat easier. Eaple : Factor each of the followig poloials. Solutio: (a) (b) 0 (c) 9 (d) (e) 8 (f) 7 (g) 0 (a) Sice the first ter is we kow that the factorig ust take the for. We kow that it will take this for because whe we ultipl the two liear ters the first ter ust be, ad the ol wa to get that to show up is to ultipl b. Therefore, the first ter i each factor ust be a. To fiish this we just eed to deterie the two ubers that eed to go i the blak spots. We ca arrow dow the possibilities cosiderabl. Upo ultiplig the two factors out, these two ubers will eed to ultipl out to get -. I other words these two ubers ust be factors of -. Here are all the possible was to factor - usig ol itegers. Now, we ca just plug these i oe after aother ad ultipl out util we get the correct pair. However, there is aother trick that we ca use here to help us out. The correct pair 9

10 of ubers ust add to get the coefficiet of the ter. So, i this case the third pair of ad will add to + ad so that is the pair we are after. factors Here is the factored for of the poloial. Agai, we ca alwas check that we got the correct aswer b doig a quick ultiplicatio. Note that the ethod we used here will ol work if the coefficiet of the ter is oe. If it is athig else, this will ot work ad we will be back to trial ad error to get the correct factorig for. (b) 0 Let s write dow the iitial for agai, 0 Now, we eed two ubers that ultipl to get ad add to get -0. It looks like - ad - are the right factors. So the factored for of this poloial is, 0 (c) 9 Let s start with the iitial for, 9 This tie we eed two ubers that ultipl to get 9 ad add to get. I this case ad will be the correct pair of ubers. Do ot forget that the two ubers ca be the sae uber o occasio as the are here. Here is the factored for for this poloial. 9 Note as well that we further siplified the factorig to ackowledge that it is a perfect square. You should alwas do this whe it happes. 0

11 (d) Oce agai, here is the iitial for, Note that, we eed two ubers that ultipl to get ad add to get. There are t such two itegers, so this quadratic is ot factorable. This will happe o occasio. (e) 8 Here, we o loger have a coefficiet of o the ter, however, we ca still ake a guess as to the iitial for of the factorig. Sice the coefficiet of the ter is a ad there are ol two positive factors of there is reall ol oe possibilit for the iitial for of the factorig. 8 Sice the ol wa to get a is to ultipl a ad a these ust be the first two ters. However, fidig the ubers for the two blaks will ot be as eas as the previous eaples. We will eed to start off with all the factors of At this poit the ol optio is to pick a pair plug the i ad see what happes whe we ultipl the ters out. Let s start with the fourth pair. Let s plug the ubers i ad see what we get. 0 8 Well the first ad last ters are correct, but the iddle ter is ot. So this is ot the correct factorig of the poloial. Now, flip the order ad see what we get. 8 So, we got it. We did guess correctl the first tie we just put the ito the wrog spot. Note that, i these probles do t forget to check both places for each pair to see if either will work. (f) 7 Agai the coefficiet of the ter has ol two positive factors so we ve ol got oe possible iitial for. 7

12 Net we eed all the factors of. Here the are. Do ot forget the egative factors. The are ofte the oes that we wat. I fact, upo oticig that the coefficiet of the is egative we ca be assured that we will eed oe of the two pairs of egative factors sice that will be the ol wa we will get egative coefficiet there. With soe trial ad error we ca get that the factorig of this poloial is, 7 (g) 0 Here, we have a harder proble. The coefficiet of the ter ow has ore tha oe pair of positive factors. This eas that the iitial for ust be oe of the followig possibilities: 0 0 To fill i the blaks we will eed all the factors of -. Here the are, With soe trial ad error we ca fid that the correct factorig of this poloial is, 0 Note as well that i the trial ad error phase we eed to ake sure ad plug each pair ito both possible fors ad i both possible orderigs to correctl deterie if it is the correct pair of factors or ot. We ca actuall go oe ore step here ad factor a out of the secod ter. This gives, 0 This is iportat because we could also have factored this as, 0 However, i this case we ca, fro the begiig, factor a out of the first ter to get: 0 CAUTION: Not ever poloial is factorable. Just like ot ever uber has a factor other tha or itself. A prie uber is a uber that has eactl two factors, ad itself.,, ad are eaples of prie ubers. The sae thig ca occur with poloials. If a poloial is ot factorable we sa that it is a prie poloial. Eaple:

13 Now tr the followig probles, ) Factor copletel the followig poloials: (a) (b) 8 (c) 0 (d) (e) (f) (g) (h) 8 (i) 8 (j) 8 (k) 8 (l) 0 More factorig probles: (a) 0 (b) 9 (c) 9 (d) 0 (e) (f) 8 (g) (h) 7 0 (i) (j) 8 (k) For ore practice, please check the hoework sheet.

14 R. Fractioal Epressios:. Operatios with Ratioal Epressios: * Equivalet fractios: * Ratioal Epressios: A ratioal epressio is the quotiet of two poloials. E:, * Siplifig Ratioal Epressios: NOTE: To siplif a ratioal epressio factor copletel the uerator ad deoiator the cacel coo factors. A ratioal epressio is reduced to lowest ters if all coo factors fro the uerator ad deoiator are caceled. Eaple a: Reduce to lowest ters Not reduced to lowest ters reduced to lowest ters With ratioal epressios it works eactl the sae wa. Not reduced to lowest ters reduced to lowest ters ** We have to be careful with cacelig. There are soe coo istakes that studets ofte ake with these probles. Reeber that i order to cacel a factor it ust ultipl the whole uerator ad the whole deoiator. So, the + above could cacel sice it ultiplied the whole uerator ad the whole deoiator. However, the s i the reduced for ca ot be cacelled, sice the i the uerator is ot ties the whole uerator. To see wh the s do t cacel i the reduced for above put a uber i ad see what happes. Let s plug i =. (If gets caceled) Clearl the two aswers are ot the sae uber! Note: Ol COMMON FACTORS of the uerator ad deoiator ca be caceled.

15 Eaple b: Siplif the followig: a 8,, a a, az a a z 8, b b a a * Multiplig Ratioal Epressios: ) Copletel factor each uerator ad deoiator. ) Multipl the uerators ad ultipl the deoiators. ) Siplif the result as far as possible b cacellig coo factors. Eaple : Eaple : Dividig Ratioal Epressios: ) Copletel factor each uerator ad deoiator. ) Chage to ultiplicatio. ) Ivert (flip) the secod fractio ad proceed as i ultiplicatio. Eaple :

16 * Fidig the Least Coo Deoiator (LCD): ) Copletel factor each deoiator. ) The LCD is the product of all uique factors each raised to the greatest power that appears i a factored deoiator. Eaple : ), z ) 7, z z z ), 9, 0 7 ), Hit: If opposite factors occur, do ot use both i the LCD. Istead, factor - fro oe of the opposite factors so that the factors are the idetical. E: If ou have factors like - ad -, these are called opposite factors. Notice that ou ca factor a - fro - so that the factors are idetical. Addig or Subtractig Ratioal Epressios: ) With the sae deoiator: If the deoiators are the sae, keep the sae deoiator just add uerators. The siplif if possible. Eaple : Eaple 7:

17 7 ) With differet deoiators: (LCD is NEEDED) a. Factor copletel each deoiator. b. Fid the LCD of the ratioal epressio. c. Write each ratioal epressio as a equivalet ratioal epressio whose deoiator is the LCD foud i step (b) d. Add or subtract uerators, ad write the result over the coo deoiator. e. Siplif the resultig ratioal epressio if possible. Eaple 8: a) b) c) 9 d) 9 9 e) * Cople Fractios: A cople fractio is a ratioal epressio whose uerator, deoiator, or both cotai oe or ore ratioal epressios. Eaples are b a 9 The ca be siplified b treatig the uerator ad the deoiator as separate probles. The we have a "divisio" proble.

18 For eaple, to siplif 9, we first coplete the subtractio proble cotaied i the uerator of 9 9 the etire fractio:. 9 Now we have the divisio proble: We ivert ad ultipl: 9. As before, we should ow factor i order to reduce: Now cacel the coo factor "." So our fial aswer i factored for, reduced to lowest ters is:. Reeber, ou do ot eed a coo deoiator whe ultiplig (or dividig) fractios. Steps for siplifig cople fractios: ) Siplif the uerator ad the deoiator of the cople fractio so that each is a sigle fractio. ) Perfor the idicated divisio b ultiplig the uerator of the cople fractio b the reciprocal of the deoiator of the cople fractio. ) Siplif if possible. 8

19 9 Eaple 9: Siplif the followig: a) 9 b) c) d) Solutio: a) b) First we siplif the uerator ad deoiator separatel so that each is a sigle fractio Note the LCD of the fractios that are i the uerator is ad the LCD of the fractios that are i the deoiator is.

20 0 Now, tr the followig probles:. Perfor the idicated operatios ad siplif our aswers. (a) (b) (c) (d) 0 (e) (f) 7 (g) (h) h h (i) a b b a

21 . Ratioal Epoets We have alread discussed iteger epoets i Sectio P.. We ca also defie epoets that are ratioal. We use "ratioal (i.e. fractioal) epoets" to represet radicals. * Defiitio of a : If a is a real uber ad is a positive iteger greater tha, the a The quatit a is called the th root of a. Cosider the followig eaples: is rewritte as is rewritte as Eaple : Evaluate the followig. (a) 7 (b) (c) 8 Solutio: (a) Applig the defiitio of 7 7 a a, where a 0 whe is eve. with =, (b) Applig the defiitio of (c) 8 a with = power, the result is ultiplied b -., Tr it o our ow.. Note that ol the uber is raised to the * Defiitio of a : a is rewritte as a. a = a a Notice that, i a fractioal epoet, the deoiator idetifies the root while the uerator idetifies the power.

22 Eaple : Evaluate the followig. (a) (b) (c) Solutio: (a) Usig the defiitio of ratioal epoets, 8 (b) To copute, First reeber that a egative epoet eas "flip the base:" Now covert to radical otatio: Reeber that it's usuall easier to copute the root before ou copute the power. So, as our fial aswer we have: (c), Tr it o our ow. The et thig that we should ackowledge is that all of the properties for epoets that we gave i Sectio P. are still valid for all ratioal epoets.

23 * Properties of Ratioal Epoets: The basic rules of ratioal epoets are siilar to those of iteger epoets. If ad are ratioal ubers the the followig rules hold: Rule : Eaple: (a) (b) That is, whe ou ultipl like bases, ou add the epoets. Rule : Eaple: That is, whe ou divide like bases, ou subtract the epoets. Rule : Eaple: That is, whe ou raise a power to a power, ou ultipl the epoets. ********************************************************** Rule : Eaple: 7 7 That is, egative epoets result i "flippig fractios over."

24 Rule : That is, the th power of a product is equal to the product of the th powers of the factors. Eaple: Note that a of these properties were give with ol two ters/factors but the ca be eteded out to as a ters/factors as we eed. For eaple, rule ca be eteded as follows. z z Warig: DO NOT distribute epoets across additio or subtractio! For eaple, DOES NOT EQUAL! Rule : That is, the th power of a quotiet is equal to the quotiet of the th powers. Eaple: 8 8 Warig: DO NOT distribute epoets across additio or subtractio! For eaple, DOES NOT EQUAL! *Zero epoets: Rule 7: 0 Eaple: 0 ad 0 Tr these o our ow.

25 . Siplif the followig ad write the aswer i ters of positive epoets: (a) 7 (b) 8 (c) (d) ) ( (e) 8 (f) b a (g) 7 (h) (i) 7. Epress the followig i ters of ratioal epoets. (a) (b) For ore practice, please check the hoework probles.

26 . Operatios with Radicals Radicals ad Ratioal epoets: Radicals ca be rewritte b usig epoets, ad epoets ca be rewritte b usig radicals accordig to a special rule of otatio. That is, we use "ratioal (i.e. fractioal) epoets" to represet radicals ad visa versa. Cosider the followig eaples: is rewritte as is rewritte as is rewritte as I geeral, is rewritte as. Keep i id that, i a fractioal epoet, the deoiator idetifies the root while the uerator idetifies the power. Eaple: Epress each of the followig i ters of ratioal epoets. (a) (b) ( ) (c) (d) (e) Solutio: (a) (b) ( ) (d) 0 (c) (e) 8

27 I this sectio, we will be addig, subtractig, ultiplig ad dividig algebraic epressios cotaiig radicals. *Addig or Subtractig Radical epressios: To add or subtract radical epressios we have to cobie like radicals. Like radicals: are radicals with the sae ide ad the sae radicad. Note: Whe addig or subtractig radicals, alwas check first whether a radicals ca be siplified. Eaple : Perfor the idicated operatio, the siplif the followig: (a) 8 8 (b) 7 7 (c) 7 7 Caot be siplified, sice 7 ad 7 do ot cotai like radicals. (d) 0 8 (e) 8 7. (f) 9. 9 *Multiplig or Dividig Radical epressios: To ultipl or divide epressios cotaiig radicals: ) Covert to Ratioal epoets. ) Appl the rules of ratioal epoets. Eaple : Perfor the idicated operatio, the siplif the followig: (a) Solutio: (b) (a) = a b a b (c) a b a b Covert to ratioal epoets Sae base, Add epoets Aswer i epoet for Aswer i siplified radical for 7

28 (b) Covert to ratioal epoets Sae base, Subtract epoets Aswer i epoet for Aswer i siplified radical for a b a b (c) a b a b a b a b Covert to ratioal epoets a b a b The look at the sae base ad deterie whether ou have to add or subtract epoets. a b a b a b a b Aswer i siplified for To ultipl ( )( ), we use FOIL. To ultipl ( ), we use the distributive propert. Check class otes. Eaple: Multipl ( )ad ( ) ( a b ) ad ( a b) * Ratioalizig Deoiators: The process of writig a epressio without a radical i the deoiator is called ratioalizig the deoiator. A) Ratioalizig Deoiators Havig Oe Ter:, 7, are epressios with oe radical ter i the deoiator. To ratioalize the deoiator of epressios with oe radical ter i the deoiator, ou eed to ultipl b whatever akes the deoiator a perfect square or perfect cube or a other power that ca be siplified. NOTE: Make sure ou ultipl b whatever akes the radicad the sallest possible value to be siplified. This will avoid havig to further siplif later o. Eaple : Ratioalize the deoiator. 8

29 (a) (b) (c) Solutio: (a) Multipl b a value that will create the sallest perfect square uder the radical. This will prevet the eed for additioal siplificatios. Choosig to ultipl b the deoiator. Replacig will create the sallest perfect square uder the radical i b, ratioalizes the deoiator. (b) Agai, Multipl b a value that will create the sallest perfect cube uder the radical. This will prevet the eed for additioal siplificatios. Choosig to ultipl b will create the sallest perfect cube uder the radical i the deoiator. = 8 Replacig 8 b, ratioalizes the deoiator. 9

30 (c) Sice the Deoiator is Just a Sigle Radical, Multipl the uerator ad deoiator b the deoiator. So, we eed to ultipl the uerator ad the deoiator b B) Ratioalizig Deoiators Havig Two Ters: are epressios with two radical ters i the deoiator. Whe there is ore tha oe ter i the deoiator, the process is a little trick. You will eed to ultipl the deoiator b its cojugate. The cojugate is the sae epressio as the deoiator but with the opposite sig i the iddle. Eaples of cojugates are: ad a b ad a b Eaple : Ratioalize the deoiator. (a) (b) (c) Solutio: (a) Sice the deoiator has two ters, we eed to ultipl the uerator ad deoiator b the cojugate of the deoiator. Whe ultiplig the uerators i this proble, use the distributive propert. Whe ultiplig the deoiators, use FOIL. The cojugate is. = Multipl top ad botto b the cojugate of the deoiator,. Notice that ou are ultiplig b, which does ot chage the origial epressio.,, 0

31 Notice what is happeig to the iddle ters whe ou ultipl the deoiators. The iddle ters will drop out. Also, the last ter has created a perfect square uder the square root. Note: Be sure to eclose epressios with ultiple ters i ( ). This will help ou to reeber to FOIL these epressios. (b) So, we took the origial deoiator ad chaged the sig o the secod ter ad ultiplied the uerator ad deoiator b this ew ter. B doig this we were able to eliiate the radical i the deoiator whe we the ultiplied out. (c) Agai, Multipl top ad botto b the cojugate of the deoiator,. Notice that ou do ot chage the sigs uder each radical. Multipl uerators the ultipl deoiators. (Check class otes) I geeral, the product of a epressio ad its cojugate will cotai o radical ters. ** Fial Note: Ratioalizig the deoiator a see to have o real uses, however, if ou are o a track that will take ou ito a Calculus class ou will fid that ratioalizig is useful o occasio at that level.. Perfor the idicated operatios ad siplif. State our aswers i radical otatio. (a) a 8a (b) a (c) a a. Ratioalize the deoiator i each of the followig. (a) (b) (c) (d)