Unit 1. Extending the Number System. 2 Jordan School District

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1 Uit Etedig the Number System Jord School District

2 Uit Cluster (N.RN. & N.RN.): Etedig Properties of Epoets Cluster : Etedig properties of epoets.. Defie rtiol epoets d eted the properties of iteger epoets to rtiol epoets.. Rewrite epressios, goig betwee rtiol d rdicl form VOCABULARY If the epoet o term is of the form m, where 0, the the umber is sid to hve rtiol epoet. 8 is emple of costt with rtiol epoet. Properties of Epoets (All bses re o-zero) b b = + Properties of Rtiol Epoets (All bses re o-zero) p r p r q s q s = + Emples = = b = b p q r s = p r q s = = 0 = Studets should hve this property memorized. 0 = = However: = 0 Therefore: = = p q = p q = ( ) y = y ( y) p q p p q q = y ( ) y = y m ( ) = m r p s p r q q s = ( ) = = = Jord School District

3 y m = y m m m m y = = y y m y p q = y p q p q m m b y = = y y m b b = y y 4 y = = y y 4 4 Prctice Eercises A Simplify ech epressio usig oly positive epoets.. /4. 8. / 4. ( ) ( ) /4 / 8. ( ) 6 0. / / / / y / y. / y. / k. / z 0 z 4 z / Defiitio A rdicl c lso be writte s term with rtiol epoet. For emple, is iteger d 0. I geerl, m m = where m d re itegers d 0. = where m = m The deomitor of the rtiol epoet becomes the ide of the rdicl. m clsobewrittes ( ) m 4 Jord School District

4 Rtiol Epoet Form 9 0 Rdicl Form 0 9 Prctice Eercises B Rewrite ech epressio i rdicl form /4. / / k 6. /. ( 64) 8. ( 8) 9. ( ) / Prctice Eercises C Rewrite ech epressio with rtiol epoets.. 4. ( ) 4. ( ) ( 0 ) 8 6. r. w 8. k 9. ( ) z Vocbulry For iteger greter th, if = k, the is the th root of k. A rdicl or the pricipl th root of k: k, the rdicd, is rel umber., the ide, is positive iteger greter th oe. Jord School District

5 Properties of Rdicls = b = b = b b Simplifyig Rdicls: Rdicls tht re simplified hve: o frctios left uder the rdicl. o perfect power fctors i the rdicd, k. o epoets i the rdicd, k, greter th the ide,. Vocbulry A prime umber is whole umber greter th tht is oly divisible by d itself. I other words, prime umber hs ectly two fctors: d itself. Emple: = prime 0 = ot prime Divisio Rules For Few Prime Numbers A umber is divisible by: If: Emple: The lst digit is eve 6 is (0,, 4, 6, 8) is ot 8 (+8+= d =4) Yes The sum of the digits is divisible by 8 (+8+=4 d 4 = 4 ) No The lst digit is 0 or If you double the lst digit d subtrct it from the rest of the umber d the swer is: 0 Divisible by is 809 is ot 6 (Double is 4, 6 4 = 6 d 6 = 9 ) Yes 90 (Double is 0, 90 0 = 80 d 80 = ) No 6 Jord School District

6 Simplifyig Rdicls Method : Fid Perfect Squres Uder the Rdicl. Rewrite the rdicd s fctors tht re perfect squres d fctors tht re ot perfect squres.. Rewrite the rdicl s two seprte rdicls.. Simplify the perfect squre. Method : Use Fctor Tree. Work with oly the rdicd.. Split the rdicd ito two fctors.. Split those umbers ito two fctors util the umber is prime umber. 4. Group the prime umbers ito pirs.. List the umber from ech pir oly oce outside of the rdicd. 6. Leve y upired umbers iside the rdicl. Note: If you hve more th oe pir, multiply the umbers outside of the rdicl s well s the umbers left iside. Method : Divide by Prime Numbers. Work with oly the rdicd.. Usig oly prime umbers, divide ech rdicd util the bottom umber is prime umber.. Group the prime umbers ito pirs. 4. List the umber from ech pir oly oce outside of the rdicd.. Leve y upired umbers iside the rdicl. Note: If you hve more th oe pir, multiply the umbers outside of the rdicl s well s the umbers left iside. Emple: Emple: Emple: Jord School District

7 Method 4: Use Epoet Rules Emple:. Rewrite the epoet s rtiol epoet.. Rewrite the rdicd s fctors tht re perfect squres d fctors tht re ot perfect squres.. Rewrite the perfect squre fctors with / ( ) / epoet of. ( ) / 4. Split up the fctors, givig ech the / / rtiol epoet. ( ) ( ). Simplify. ( ) / 6. Rewrite s rdicl Method 4 with Vribles: Emple:. Rewrite the epoet s rtiol epoet.. Rewrite the rdicd s two fctors. Oe with the highest epoet tht is divisible by the root d the other fctor with epoet of wht is left over.. Split up the fctors, givig ech the / ( 6 ) / rtiol epoet. ( ) / 6 / 4. Rewrite the epoets usig epoet rules. /. Simplify. 6. Rewrite s rdicl 6/ / Prctice D Simplify ech rdicl epressio.. 4 p. 80 p. 4 y 4. 6u 4 v. y 6. 64m. 4 6y r 9. 96m 8 Jord School District

8 Uit Cluster (N.RN.): Usig Properties of Rtiol d Irrtiol umbers Cluster : Etedig properties of epoets.. Properties of rtiol d irrtiol umbers (i.e. sum of rtiol umbers is rtiol, sum of rtiol d irrtiol umber is irrtiol) Number Systems Comple Numbers: ll umbers of the form + bi where d b re rel umbers i, i Rel Numbers Rtiol Numbers cosist of ll umbers tht c be writte s the rtio of two itegers, 0.,, 0.9 Itegers re the whole umbers d their opposites (-, -, -, 0,,,, ) Nturl Numbers (Coutig Numbers) Whole Numbers iclude zero d the turl umbers Imgiry Numbers: re of the form of bi where i = -i, i Irrtiol Numbers cosist of ll umbers tht cot be writte s the rtio of 4 two itegers. π,, 9 Jord School District

9 Properties of Rel Numbers Descriptio Numbers Algebr Commuttive Property + = + + b = b + You c dd or multiply rel umbers i y order without ( ) = ( ) b = b chgig the result. Associtive Property The sum or product of three or more rel umbers is the sme ( + ) + = + ( + ) ( b) c = ( bc) regrdless of the wy the umbers re grouped. Distributive Property Whe you multiply sum by umber, the result is the sme ( + 8) = ( ) + ( 8) ( b + c) = b + bc whether you dd d the multiply or whether you multiply ( + 8) = ( ) + ( 8) ( b + c) = b + c ech term by the umber d the dd the products. Additive Idetity Property The sum of umber d 0, the dditive idetity, is the origil umber. + 0 = + 0 = 0 + = Multiplictive Idetity Property The product of umber d, the multiplictive idetity, is the origil umber. Additive Iverse Property The sum of umber d its opposite, or dditive iverse, is 0. Multiplictive Iverse Property The product of o-zero umber d its reciprocl, or multiplictive iverse Closure Property The sum or product of y two = = = + ( ) = 0 + ( ) = 0 8 = 8 + = rel umbers is rel umber. ( ) 6 = =, 0 + b b 0 Jord School District

Why c t I come i????? Sorry we re CLOSED set! Closure Whe opertio is eecuted o the members of set, the result is gurteed to be i the set.

Therefore, Emple: ( 6)( ) = 4 itegers re closed uder multiplictio. Subtrctio: If oe iteger is subtrcted from other, the differece is iteger. Therefore, itegers re closed uder subtrctio.

Emple: You Decide. Wht umber systems re closed uder dditio? Justify your coclusios usig the method of your choice.. Wht umber systems re closed uder multiplictio?

10 Why c t I come i????? Sorry we re CLOSED set! Closure Whe opertio is eecuted o the members of set, the result is gurteed to be i the set. Additio: If two itegers re dded together, Emple: + = the sum is iteger. Therefore, itegers re closed uder dditio. Multiplictio: If two itegers re multiplied together, the product is iteger. Therefore, Emple: ( 6)( ) = 4 itegers re closed uder multiplictio. Subtrctio: If oe iteger is subtrcted from other, the differece is iteger. Therefore, itegers re closed uder subtrctio. Emple: ( 6) = 4 Divisio: If oe iteger is divided by other 0 ( ) = closed iteger, the quotiet my or my ot be iteger. Therefore, itegers re ot closed ( ) 0 = ot closed uder divisio. Emple: You Decide. Wht umber systems re closed uder dditio? Justify your coclusios usig the method of your choice.. Wht umber systems re closed uder multiplictio? Justify your coclusios usig the method of your choice.. Wht umber systems re closed uder subtrctio? Justify your coclusios usig the method of your choice. 4. Wht umber systems re closed uder divisio? Justify your coclusios usig the method of your choice. Jord School District

11 Vocbulry For iteger greter th, if = k, the is the th root of k. A rdicl or the pricipl th root of k: k, the rdicd, is rel umber., the ide, is positive iteger greter th oe. Properties of Rdicls = b = b = b b Simplifyig Rdicls: Rdicls tht re simplified hve: o frctios left uder the rdicl. o perfect power fctors i the rdicd, k. o epoets i the rdicd, k, greter th the ide,. Vocbulry A prime umber is whole umber greter th tht is oly divisible by d itself. I other words, prime umber hs ectly two fctors: d itself. Emple: = prime 0 = ot prime Divisio Rules For Few Prime Numbers A umber is divisible by: If: Emple: The lst digit is eve 6 is divisible by (0,, 4, 6, 8) is ot divisible by 8 (+8+= d =4) Yes The sum of the digits is divisible by 4 = 4 8 (+8+=4 d ) No The lst digit is 0 or is divisible by 809 is ot divisible by If you double the lst digit 6 (Double is 4, d d subtrct it from the rest 6 = 9 ) Yes of the umber d the swer is: 90 (Double is 0, d 0 Divisible by 80 = ) No Jord School District

12 Simplifyig Rdicls Method : Fid Perfect Squres Uder the Rdicl 4. Rewrite the rdicd s fctors tht re perfect squres d fctors tht re ot perfect squres.. Rewrite the rdicl s two seprte rdicls. 6. Simplify the perfect squre. Method : Use Fctor Tree. Work with oly the rdicd. 8. Split the rdicd ito two fctors. 9. Split those umbers ito two fctors util the umber is prime umber. 0. Group the prime umbers ito pirs.. List the umber from ech pir oly oce outside of the rdicd.. Leve y upired umbers iside the rdicl. Note: If you hve more th oe pir, multiply the umbers outside of the rdicl s well s the umbers left iside. Method : Divide by Prime Numbers 6. Work with oly the rdicd.. Usig oly prime umbers, divide ech rdicd util the bottom umber is prime umber. 8. Group the prime umbers ito pirs. 9. List the umber from ech pir oly oce outside of the rdicd. 0. Leve y upired umbers iside the rdicl. Note: If you hve more th oe pir, multiply the umbers outside of the rdicl s well s the umbers left iside. Emple: Emple: Emple: Jord School District

13 Method 4: Use Epoet Rules. Rewrite the epoet s rtiol epoet. 8. Rewrite the rdicd s fctors tht re perfect squres d fctors tht re ot perfect squres. 9. Rewrite the perfect squre fctors with Emple: / ( ) / epoet of. ( ) / 0. Split up the fctors, givig ech the / / rtiol epoet. ( ) ( ). Simplify. ( ) /. Rewrite s rdicl Method 4 with Vribles: Emple:. Rewrite the epoet s rtiol epoet. 8. Rewrite the rdicd s two fctors. Oe with the highest epoet tht is divisible by the root d the other fctor with epoet of wht is left over. 9. Split up the fctors, givig ech the / ( 6 ) / rtiol epoet. ( ) / 6 / 0. Rewrite the epoets usig epoet rules. /. Simplify.. Rewrite s rdicl 6/ / Addig d Subtrctig Rdicls To dd or subtrct rdicls, simplify first if possible, d the dd or subtrct like rdicls. Emple:. They both hve the sme term uder the rdicl so they re like terms. +. Add the coefficiets of the rdicls. ( + ) Emple:. They both hve the sme term uder 4 the rdicl so they re like terms.. Subtrct the coefficiets of the rdicls. (4 ) ( ) 4 Jord School District

14 Emple:. They re ot like terms, but oe of + them c be simplified.. Rewrite the umber uder the rdicl. +. Use the properties of rdicls to write the fctors s two rdicls is perfect squre d the squre root of it is.. Multiply the coefficiets of the secod rdicl. 6. Now they re like terms, dd the coefficiets ( + 0). Noe of them re like terms. Simplify if you c. Emple: Fctor ech umber iside the rdicl Use the properties of rdicls to simplify d 9 re perfect squres; their squre roots re d.. Multiply the umbers outside of the rdicl Oly the terms with re like (0 9) + terms.. Simplify. +. They re ot like terms, but they c be simplified.. Rewrite the epressios uder the rdicl.. Use properties of rdicls to rewrite the epressios. 4. The cube root of 8 is d the cube root of is.. Multiply the coefficiets of the lst rdicl. 6. Add or subtrct the coefficiets of the like terms.. Simplify. Emple: ( ) + 6 Jord School District

15 . Emple: They re ot like terms, but oe of them c be simplified. 4. Rewrite the epressio uder the rdicl. 8. Use properties of rdicls to rewrite the epressio d re perfect cubes. The cube root of 8 is d is.. Multiply the coefficiets of the first rdicl Now they re like terms, dd the coefficiets of ech. ( 4 ). Simplify. Multiplyig Rdicls Multiplyig rdicls with the sme ide. Multiply the coefficiets d multiply the umbers uder the rdicd.. If possible, simplify. This is lredy simplified. Emple: 6 Emple: ( ) ( + )( 6 ). Use the distributive method to multiply. ( 6 ) + ( ) ( ). Use properties of rdicls to simplify Simplify y rdicls Combied like terms if possible Emple: ( )( 4). Use the distributive method to multiply. + ( 4) + ( ) + ( )( 4). Use properties of rdicls to simplify Simplify d combie like terms. + ( 4 ) The squre root of is Combie like terms. 9 6 Jord School District

16 Emple: ( + )( ). Use the distributive method to multiply. + ( ) + ( ) + ( ). Use properties of rdicls to simplify Simplify d combie like terms. 4 + ( + ) 9 4. The squre root of 4 is. 9. Simplify Multiplyig Rdicls with Differet Idices Note: I order to multiply rdicls with differet idices the rdicds must be the sme. Emple:. Rewrite ech rdicl usig rtiol epoets.. Use properties of epoets to simplify.. Combie the frctios by fidig commo deomitor.. Rewrite ech rdicl usig rtiol epoets.. Use properties of epoets to simplify.. Combie the frctios by fidig commo deomitor.. Rewrite the ier rdicl usig rtiol epoets.. Rewrite the outer rdicl usig rtiol epoets. + 0 Emple: Emple:. Use properties of epoets to simplify. 4. Simplify by multiplyig frctios. 6 Jord School District

17 Prctice Eercises A Add or Subtrct Prctice Eercises B Multiply d simplify the result ( + ). 4 ( ) 6. ( ). ( )( + ) 8. ( + )( ) 9. ( 4 )( + ) y y. 8 z 8 Jord School District

18 You Decide: 4. Add: +. C you write the result s the rtio of two umbers? (Use your grphig clcultor to chge the sum from deciml to frctio by pushig the mth butto d select FRAC). Add: +. C you write the result s the rtio of two umbers?. Add: +.. C you write the result s the rtio of two umbers? 4. Add:. +.. C you write the result s the rtio of two umbers?. Add: +. C you write the result s the rtio of two umbers? 6. Add: + π. C you write the result s the rtio of two umbers?. Write rule bsed o your observtios with ddig rtiol d irrtiol umbers. 9 Jord School District

19 Uit Cluster 4 (A.APR.): Polyomils Cluster 4: Perform rithmetic opertios o polyomils.4. Polyomils re closed uder dditio, subtrctio, d multiplictio.4. Add, subtrct, d multiply polyomils (NO DIVISION) VOCABULARY A term tht does ot hve vrible is clled costt. For emple the umber is costt becuse it does ot hve vrible ttched d will lwys hve the vlue of. A costt or vrible or product of costt d vrible is clled term. For emple re ll terms.,, or Terms with the sme vrible to the sme power re like terms. d re like terms. A epressio formed by ddig fiite umber of ulike terms is clled polyomil. The vribles c oly be rised to positive iteger epoets is polyomil, while + is ot polyomil. NOTE: There re o squre roots of vribles, o frctiol powers, d o vribles i the deomitor of y frctios. 4 A polyomil with oly oe term is clled moomil( 6 ). A polyomil with two terms is clled biomil ( + ). A polyomil with three terms is clled triomil( + ). Polyomils re i stdrd (geerl) form whe writte with epoets i descedig order d the 4 costt term lst. For emple + + is i stdrd form. The epoet of term gives you the degree of the term. The term hs degree two. For polyomil, the vlue of the lrgest epoet is the degree of the whole polyomil. The polyomil hs degree 4. The umber prt of term is clled the coefficiet whe the term cotis vrible d umber. 6 hs coefficiet of 6 d hs coefficiet of -. The ledig coefficiet is the coefficiet of the first term whe the polyomil is writte i stdrd 4 form. is the ledig coefficiet of + +. Degree Geerl Polyomil: f ( ) = Ledig Coefficiet Ledig Term 0 Jord School District Costt

20 CLASSIFICATIONS OF POLYNOMIALS Nme Form Degree Emple Zero f ( ) = 0 Noe f ( ) = 0 Costt f ( ) =, 0 0 f ( ) = Lier f ( ) = + b f ( ) = + Qudrtic Cubic ( ) = + + f b c ( ) = f b c d f ( ) = f ( ) = Prctice Eercises A: Determie which of the followig re polyomil fuctios. If the fuctio is polyomil, stte the degree d ledig coefficiet. If it is ot, epli why.. f ( ) = +. f ( ) = 9 +. f ( ) = f ( ) =. 6. f ( ) = f ( ) = 4 Opertios of Polyomils Additio/Subtrctio: Combie like terms. Emple : Horizotl Method Verticl Method ( + 4 ) + ( + + ) = ( + ) + ( + ) + ( 4 ) + ( + ) = Emple : Horizotl Method ( ) ( + + ) = = Verticl Method ( ) Jord School District

21 Multiplictio: Multiply by moomil Emple : Emple 4: ( + ) ( ) ( ) ( ) 6 = = + ( ) = Multiplictio: Multiply two biomils ( )( + 9) Distributive (FOIL) Method ( )( + 9) = ( + 9) ( + 9) = * combielike terms = Bo Method *combie terms o the digols of the ushded boes(top right to lower left) Verticl Method = Multiplictio: Multiply biomil d triomil ( + )( 6 ) Distributive Method ( + )( 6 ) = ( 6 ) + ( 6 ) = ( 4 0) + ( 8 ) = * combie liketerms = Bo Method *combie terms o the digols of the ushded boes(top right to lower left) Verticl Method = + 4 Jord School District

22 Prctice Eercises B: Perform the required opertios. Write your swers i stdrd form d determie if the result is polyomil.. ( + ) + ( + ). ( ) ( + + ). ( 4 + ) ( + ) 4. ( y + y ) + ( y + y + 4). ( + ) 6. y ( y + y 4). u ( 4u ) 8. ( )( ) 9. ( + )( ) 0. ( )( + ). ( + )( 4 + ). ( y)( + y). ( + ) 4. ( ). ( ) 6. ( y)( + y). ( + )( + 4) 8. ( + )( ) 9. ( + )( + + ) 0. ( + ) + ( + + ) YOU DECIDE Are polyomils closed uder dditio, subtrctio, multiplictio? Justify your coclusio usig the method of your choice. Jord School District

SM2H. Unit 2 Polynomials, Exponents, Radicals & Complex Numbers Notes. 3.1 Number Theory

SM2H. Unit 2 Polynomials, Exponents, Radicals & Complex Numbers Notes. 3.1 Number Theory SMH Uit Polyomils, Epoets, Rdicls & Comple Numbers Notes.1 Number Theory .1 Addig, Subtrctig, d Multiplyig Polyomils Notes Moomil: A epressio tht is umber, vrible, or umbers d vribles multiplied together.