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. Moomils oly hve vribles with whole umber epoets d ever hve vribles i the deomitor of frctio or vribles uder roots. Moomils: yz 1 b,, w,,, y 8 4 Not Moomils: 1 4,,, z 1 1 Costt: A moomil tht cotis o vribles, like or 1. Coefficiet: The umericl prt of moomil (the umber beig multiplied by the vribles.) Polyomil: A moomil or severl moomils joied by + or sigs. Terms: The moomils tht mke up polyomil. Terms re seprted by or sigs. Like Terms: Terms whose vribles d epoets re ectly the sme. Biomil: A polyomil with two ulike terms. Triomil: A polyomil with three ulike terms. Addig d Subtrctig Polyomils To dd or subtrct polyomils, combie like terms. Add or subtrct the coefficiets. The vribles d epoets do ot chge. Remember to subtrct everythig iside the pretheses fter mius sig. Subtrct mes dd the opposite, so chge the mius sig to plus sig d the chge the sigs of ll the terms iside the pretheses. Emples: Simplify ech epressio. ) 7 b) 4 1 6 c) w w 4w w d) 6 4 i) 6m m 4m m m 7m j) k k k 4k 8
Multiplyig Polyomils To multiply two polyomils, distribute ech term of oe polyomil to ech term of the other polyomil. The combie y like terms. Whe you re multiplyig two biomils, this is sometimes clled the FOIL Method becuse you multiply F the first terms, O the outside terms, I the iside terms, d L the lst terms. Emples: Multiply. b) mm 8 ) y 7 y y 11 c) 1 d) u 1u 4 e) f) g) 6 7 h) 4 7 8 Perimeter Perimeter = sum of ll the sides The mesure of the perimeter of trigle is s + 6. It is kow tht two of the sides of the trigle hve mesures of s - d s + 14. Fid the legth of the third side.
. Rules of Epoets The followig properties re true for ll rel umbers d b d ll itegers m d, 0 provided tht o deomitors re 0 d tht 0 is ot cosidered. 1 s epoet: 1 1 1 e.g.) 1 7 7, π π, 10 10 0 s epoet: 1, 7 1, 1 0 0 0 1 e.g.) 0 8 The Product Rule: m m e.g.) 7 The Quotiet Rule: m m e.g.) m The Power Rule: m 10 e.g.) Risig product to power: b b e.g.) 4 4 4 4 k k 16k Risig quotiet to power: b b e.g.) p p p q q q 6 Negtive epoets: 1 e.g.) 1 7, 7 y 4 y 4 1 e.g.) 1 9 b bc, 9 c d d b b b e.g.) v v v v 8 Rtiol epoets: d d e.g.)
To simplify epressio cotiig powers mes to rewrite the epressio without pretheses or egtive epoets. Emples: Simplify the followig epressios. ) m 4 m b) b b c) 7 r r 9 d) p p 7 e) 4y 6 7 y f) 4 g) 4 h) y y 4 1 1 i) j) 6 8 9 9 k) 4 1 y l) y 4 y m) 7 y ) 1 o) y p) yz
. Rtiol Epoets If is positive iteger greter th 1 d is rel umber the 1. The deomitor of the epoet tells you wht type of root to tke. Emples: Write equivlet epressio usig rdicl ottio d, if possible, simplify. ) 1 b) y z 1 64 c) 16 10 d) 6 1 e) 14 f) 14 Emples: Write equivlet epressio usig epoetil ottio. ) 7 y b) 4 b 7 c) z d) z e) y z Positive Rtiol Epoets If m d re positive itegers (where 1) d eists, the m m. 8 8 4 or 8 8 64 4 e.g.) m Emples: Write equivlet epressio usig rdicl ottio d simplify. ) 6 t b) 9 c) 64 d) 4 e) 4 Emples: Write equivlet epressio usig epoetil ottio. ) b) 7 9 c) 6 d) 6 e) 4 m
Negtive Rtiol Epoets m For y rtiol umber m, d y ozero rel umber, The sig of the bse is ot ffected by the sig of the epoet. m 1 m. Emples: Write equivlet epressio usig positive epoets d, if possible, simplify. ) 1 49 b) m c) 7 Lws of Epoets: The lws of epoets pply to rtiol epoets s well s iteger epoets. Emples: Use the lws of epoets to simplify. 7 1 ) b) 4 c) 19 d) 1 e) y y f) 4 7 6 7 z z 4 g) 4 1 6 y y 1 1 y h) y To Simplify Rdicl Epressios usig the Rules of Epoets: 1. Covert rdicl epressios to epoetil epressios.. Use rithmetic d the lws of epoets to simplify.. Covert bck to rdicl ottio s eeded. Emples: Use rtiol epoets to simplify. Do ot use epoets tht re frctios i the fil swer. ) 8 4 9 z b) 4 bc c) 4 d) y 6 9 y e) k k 7 f) 8 4 6 m m g) 4 h)
. Simplifyig Rdicl Epressios Squre Root: A umber tht you squre (multiply by itself) to ed up with is clled squre root of. I symbols, k if k. Rdicl Sig: The symbol. The rdicl sig is used to idicte the pricipl (positive) squre root of the umber over which it ppers. Rdicd: The umber uder the rdicl sig. Perfect squres: Numbers tht re the squres of rtiol umbers. Emples: 1 16 1, 4, 9, 81,,, etc. 6 Emples: Simplify ech of the followig: ) 196 b) 6 c) 49 81 d) 4 y e) 14 z th Root: A umber tht you rise to the th power (multiply by itself times) to ed up with is clled th root of. I symbols, k if k. Ide: I the epressio, is clled the ide. It tells you wht root to tke. Emples: Simplify ech epressio, if possible. ) 1 b) 4 81 c) d) 6 8y Simplified Rdicl Epressios: No perfect th power fctors i the rdicd No epoets i the rdicd bigger th the ide No frctios i the rdicd The ide is s smll s possible To Simplify Rdicl Epressio with Ide by Fctorig: 1. Write the rdicd s the product of perfect th powers d fctors tht re ot perfect th powers.. Rewrite the epressio s the product of seprte th roots.. Simplify ech epressio cotiig the th root of perfect th power. To Simplify Rdicl Epressio with Ide Usig Fctor Tree: 1. Mke fctor tree. Split the rdicd ito its prime fctors.. Circle groups of ideticl fctors.. List the umber or vrible from ech group oly oce outside the rdicl. 4. Leve fctors tht re ot prt of group uder the rdicl.. Multiply the fctors outside of the rdicl together. Do the sme for the fctors uder the rdicl.
Emples: Simplify ech epressio. ) 1 b) 40 c) 7 d) 0 y e) y 00 y f) 4 g) 7 40 h) t 7 u 9 i) m 40m 6 j) 4 40 k) 6 9 4 y z l) 7 14 pr p q r Opertios with Rdicls Addig d Subtrctig Rdicls: 1. Simplify ech rdicl completely.. Combie like rdicls. Whe you dd or subtrct rdicls, you c oly combie rdicls tht hve the sme ide d the sme rdicd. The rdicl itself (the root) does ot chge. You simply dd or subtrct the coefficiets. Like Rdicls: Rdicls with the sme ide d the sme rdicd. Emples: Determie whether the followig re like rdicls. If they re ot, epli why ot. ) d b) 4 d c) d
Emples: Add or subtrct. ) 7 b) 4 11 8 11 c) 10 6 8 6 d) 0 0 4 e) 0 4 00 6 1 f) 4 16 Do t mke the followig mistkes: 7 9 16 4 m m Multiplyig Rdicls The Product Rule for Rdicls: For y rel umbers d b, b b. Cutio: The product rule does t work if you re tryig to multiply the eve roots of egtive umbers, becuse those roots re ot rel umbers. For emple, 8 16. Re-write the rdicl i terms of i first, d the multiply. For emple, 8 i i 8 i 16 ( 1) 16 4 Cutio: The product oly pplies whe the rdicls hve the sme ide: 4 6 1 0. Emples: Multiply. ) 7 b) 8 c) 7 1 d) e) 8 f) 11 g) 9 h) 10 6
.4 Rdicl Epressios, Multiply d Divide (Rtiolizig the Deomitor) Questio: C you dd d subtrct rdicls the sme wy you multiply d divide them? e.g.) Sice b b, does b b? NO!!!!!!!!!! Do t mke the followig mistkes: 4 y y Multiplyig Rdicl Epressios: Use the Product Property. Use the Distributive Property d FOIL to multiply rdicl epressios with more th oe term. Emples: Multiply. ) 0 b) 6 c) 6 7 1 d) 4 e) 4 1 f) Dividig Rdicls The Quotiet Rule for Rdicls For y rel umbers d b, where b 0,. b b Emples: Simplify. ) 9 b) 7 c) m 16 d) 0y 6 8 11
Emples: Divide d, if possible, simplify. ) 7 b) 0 c) 7 48 6y y 4 8 Rtiolizig Deomitors with Oe Term: Rtiolizig the deomitor mes to write the epressio s equivlet epressio but without rdicl i the deomitor. To do this, multiply by 1 uder the rdicl or multiply by 1 outside the rdicl to mke the deomitor perfect power. Emples: Rtiolize ech deomitor. ) b) c) d) 11 Rtiolizig Deomitors with Two Terms: To do this, multiply by 1 uder the rdicl or multiply by 1 outside the rdicl to mke the deomitor perfect power. However, sice the deomitor ow hs two terms, we will hve to multiply by the cojugte of the deomitor. Cojugte of biomil Rdicl Epressio: Cojugtes hve the sme first term, with the secod terms beig opposites. For emple, these two epressios re cojugtes: d. Wht hppes whe you multiply these cojugtes together?
Emples: Fid the cojugte of ech umber. ) 4 b) 7 c) 1 Emples: Rtiolize ech deomitor by multiplyig by the cojugte. ) 4 b) 8 c)
. Simplifyig with Comple Numbers Imgiry Numbers For ceturies, mthemticis kept ruig ito problems tht required them to tke the squre roots of egtive umbers i the process of fidig solutio. Noe of the umbers tht mthemticis were used to delig with (the rel umbers) could be multiplied by themselves to give egtive. These squre roots of egtive umbers were ew type of umber. The Frech mthemtici Reé Descrtes med these umbers imgiry umbers i 167. Ufortutely, the me imgiry mkes it soud like imgiry umbers do t eist. They do eist, but they seem strge to us becuse most of us do t use them i dy-to-dy life, so we hve hrd time visulizig wht they me. However, imgiry umbers re etremely useful (especilly i electricl egieerig) d mke my of the techologies we use tody (rdio, electricl circuits) possible. The umber i: i is the umber whose squre is 1. Tht is, i 1 d i 1. We defie the squre root of egtive umber s follows: 1 1 i or i. Emples: Epress i terms of i. ) 64 b) 1 c) 49 d) 18 Imgiry Number: A umber tht c be writte i the form bi, where d b re rel umbers d b 0. Ay umber with i i it is imgiry.
Comple Number: A umber tht c be writte i the form bi, where d b re rel umbers. ( or b or both c be 0.) The set of comple umbers is the set cotiig ll of the rel umbers d ll of the imgiry umbers. Addig or Subtrctig Comple Numbers i cts like y other vrible i dditio d subtrctio problems. Distribute y egtive sigs d combie like terms (dd or subtrct the rel prts d dd or subtrct the imgiry prts). Write your swer with the rel prt first, the the imgiry prt. Emples: Add or subtrct d simplify. ) i 1 i b) 4i i c) 7i 6 d) i1 i Multiplyig Comple Numbers Multiplyig Comple Numbers: To multiply imgiry umbers, first write y squre roots of egtive umbers i terms of i. Multiply s usul by distributig, FOILig, d usig epoet rules. Tret i like y other vrible. Use the fct tht i 1. Aywhere you see i, chge it to 1. o 8i 81 8 o i 1 Emples: Multiply d simplify. If the swer is imgiry, write it i the form bi. ) 9 4 b) c) i 7i d) ii e) i i f) 7 i9 8i g) i h) 4i 4i Simplify Power of i : Epress the give power of i i terms of powers of Emples: Simplify ech epressio. ) i b) i c) i, d use the fct tht i 1. 7 i d) i 47
.6 Dividig Comple Numbers Cojugte of Comple Number: The comple cojugte of comple umber bi is bi. bi bi b. Emples: Fid the cojugte of ech umber. ) 4i b) 1 i c) i Dividig Comple Numbers: Multiply both the umertor d the deomitor by the comple cojugte of the deomitor. Emples: Divide d simplify to the form bi. ) 7 i b) 6 i i c) 9i 76i d) i 4 i