Secondary Math 2 Honors. Unit 2 Polynomials, Exponents, Radicals & Complex Numbers

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1 Secodr Mth Hoors Uit Polomils, Epoets, Rdicls & Comple Numbers. Addig, Subtrctig, d Multiplig Polomils Notes Moomil: A epressio tht is umber, vrible, or umbers d vribles multiplied together. Moomils ol hve vribles with whole umber epoets d ever hve vribles i the deomitor of frctio or vribles uder roots. Moomils: z b,, w,,, 8 4 Not Moomils: 4,,, z Costt: A moomil tht cotis o vribles, like or. Coefficiet: The umericl prt of moomil (the umber beig multiplied b the vribles.) Polomil: A moomil or severl moomils joied b + or sigs. Terms: The moomils tht mke up polomil. Terms re seprted b + or sigs. Like Terms: Terms whose vribles d epoets re ectl the sme. Biomil: A polomil with two ulike terms. Triomil: A polomil with three ulike terms. Emples: Decide whether ech epressio is polomil. If it is t, epli wh ot. ) b) 4 c) m + d) 6c + c e) 6z z + f) 7 g) 8 h) +

2 Addig d Subtrctig Polomils To dd or subtrct polomils, combie like terms. Add or subtrct the coefficiets. The vribles d epoets do ot chge. Remember to subtrct everthig 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: Simplif ech epressio. ) ( ) + ( 7 ) b) ( 4 + ) + ( + 6) c) ( w + w) ( 4w + w) d) ( 6 + ) ( 4 + ) i) ( 6m + m) ( 4m m) + ( m 7m) j) ( k+ ) + ( k k) ( 4k + 8) Multiplig Polomils To multipl two polomils, distribute ech term of oe polomil to ech term of the other polomil. The combie like terms. Whe ou re multiplig two biomils, this is sometimes clled the FOIL Method becuse ou multipl F the first terms, O the outside terms, I the iside terms, d L the lst terms. Emples: Multipl. + b) ( m+ )( m 8) ) ( 7 ) c) ( + )( ) d) ( t 4)( t+ 9)

3 e) ( u )( u 4) + f) ( ) g) ( + )( ) h) ( + )( ) k) ( )( 6+ 7) l) ( )( + 8). Rules of Epoets The followig properties re true for ll rel umbers d b d ll itegers m d, provided tht o 0 deomitors re 0 d tht 0 is ot cosidered. s epoet: = e.g.) ( ) 7 = 7, π = π, 0 = 0 0 s epoet: = e.g.) ( ) 0 8 =, 7 =, = The Product Rule: m m = + e.g.) = = + 7 The Quotiet Rule: m m = e.g.) = = m The Power Rule: ( ) m ( )( ) 0 = e.g.) ( ) = = Risig product to power: ( ) b = b e.g.) ( ) k = k = 6k Risig quotiet to power: = b b e.g.) p p p = = q q ( q ) 6

4 Negtive epoets: = e.g.) 7, 7 4 = = 4 = e.g.) 9 b bc =, = 9 c d d b b = = b e.g.) v v v = = = v 8 To simplif epressio cotiig powers mes to rewrite the epressio without pretheses or egtive epoets. Emples: Simplif the followig epressios. ) m 4 m b) ( b)( b ) c) 7 r r 9 d) p p 7 e) f) ( ) 4 g) 4 h) 4 i) 6 j) k) l) 4 4 m) ( ) 7 ) ( ) o) ( ) p) z

5 . Rtiol Epoets If is positive iteger greter th d is rel umber the =. The deomitor of the epoet tells ou wht tpe of root to tke. Emples: Write equivlet epressio usig rdicl ottio d, if possible, simplif. ) b) 64 c) ( z ) 6 0 d) ( 6 ) e) 4 f) ( ) 4 Emples: Write equivlet epressio usig epoetil ottio. b ) 7 b) 4 c) z d) z e) 7 z Positive Rtiol Epoets = = If m d re positive itegers (where ) d eists, the ( ) m m. 8 = 8 = = 4 or 8 = 8 = 64 = 4 e.g.) ( ) m Emples: Write equivlet epressio usig rdicl ottio d simplif. 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

6 Negtive Rtiol Epoets m For rtiol umber m, d ozero rel umber, The sig of the bse is ot ffected b the sig of the epoet. m = m. Emples: Write equivlet epressio usig positive epoets d, if possible, simplif. ) 49 b) ( m) c) 7 Lws of Epoets: The lws of epoets ppl to rtiol epoets s well s iteger epoets. Emples: Use the lws of epoets to simplif. 7 ) b) 4 c) ( 9 ) d) e) f) z z 4 g) 4 6 ( h) ) To Simplif Rdicl Epressios usig the Rules of Epoets:. Covert rdicl epressios to epoetil epressios.. Use rithmetic d the lws of epoets to simplif.. Covert bck to rdicl ottio s eeded. Emples: Use rtiol epoets to simplif. Do ot use epoets tht re frctios i the fil swer. ) z b) ( 4 ) bc c) 4 d) 6 9 e) k k 7 f) m m g) 4 h)

7 .4 Simplifig Rdicl Epressios Squre Root: A umber tht ou squre (multipl b itself) to ed up with is clled squre root of. I smbols, k = if k =. Rdicl Sig: The smbol. 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: 6, 4, 9, 8,,, etc. 6 Emples: Simplif ech of the followig: ) 96 b) 6 c) 49 8 d) 4 e) 4 z th Root: A umber tht ou rise to the th power (multipl b itself times) to ed up with is clled th root of. I smbols, k = if k =. Ide: I the epressio, is clled the ide. It tells ou wht root to tke. Emples: Simplif ech epressio, if possible. ) b) 4 8 c) d) 6 8 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

8 To Simplif Rdicl Epressio with Ide b Fctorig:. 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.. Simplif ech epressio cotiig the th root of perfect th power. To Simplif Rdicl Epressio with Ide Usig Fctor Tree:. Mke fctor tree. Split the rdicd ito its prime fctors.. Circle groups of ideticl fctors.. List the umber or vrible from ech group ol oce outside the rdicl. 4. Leve fctors tht re ot prt of group uder the rdicl.. Multipl the fctors outside of the rdicl together. Do the sme for the fctors uder the rdicl. Emples: Simplif ech epressio. ) b) 40 c) 7 d) 0 e) 00 f) 4 g) 7 40 h) t 7 u 9 i) m 40m 6 j) 4 40 k) z l) 7 4 pr p q r

9 . Opertios with Rdicls Addig d Subtrctig Rdicls:. Simplif ech rdicl completel.. Combie like rdicls. Whe ou dd or subtrct rdicls, ou c ol combie rdicls tht hve the sme ide d the sme rdicd. The rdicl itself (the root) does ot chge. You simpl 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 the re ot, epli wh ot. ) d b) 4 d c) d Emples: Add or subtrct. ) 7 b) c) d) e) f) Do t mke the followig mistkes: m m Multiplig Rdicls The Product Rule for Rdicls: For rel umbers d b, b= b. Cutio: The product rule does t work if ou re trig to multipl the eve roots of egtive umbers, becuse those roots re ot rel umbers. For emple, 8 6. Re-write the rdicl i terms of i first, d the multipl. For emple, 8 = i i 8 = i 6 = ( ) 6 = 4 Cutio: The product ol pplies whe the rdicls hve the sme ide:

10 Emples: Multipl. ) 7 b) 8 c) 7 d) e) ( ) 8 f) ( ) g) 9 h) 0 6 Questio: C ou dd d subtrct rdicls the sme w ou multipl d divide them? e.g.) Sice b = b, does + b = + b? NO!!!!!!!!!! Do t mke the followig mistkes: 4 ( ) + + Multiplig Rdicl Epressios: Use the Product Propert. Use the Distributive Propert d FOIL to multipl rdicl epressios with more th oe term. Emples: Multipl. ) ( + 0 ) b) ( 6 ) c) ( 6)( 7+ ) d) ( 4 )( + ) e) ( 4 ) f) ( + )( )

11 .6 Dividig Rdicls The Quotiet Rule for Rdicls For rel umbers d b, where b 0, =. b b Emples: Simplif. ) 9 b) 7 c) m 6 d) Emples: Divide d, if possible, simplif. ) 7 b) 0 c) 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, multipl b uder the rdicl or multipl b outside the rdicl to mke the deomitor perfect power. Emples: Rtiolize ech deomitor. ) b) c) d)

12 Rtiolizig Deomitors with Two Terms: To do this, multipl b uder the rdicl or multipl b outside the rdicl to mke the deomitor perfect power. However, sice the deomitor ow hs two terms, we will hve to multipl b 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 ou multipl these cojugtes together? ( )( + ) = Emples: Fid the cojugte of ech umber. ) 4+ b) 7 c) Emples: Rtiolize ech deomitor b multiplig b the cojugte. ) 4 + b) 8 c) +

13 .7 Simplifig with Comple Numbers Imgir 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 b themselves to give egtive. These squre roots of egtive umbers were ew tpe of umber. The Frech mthemtici Reé Descrtes med these umbers imgir umbers i 67. Ufortutel, the me imgir mkes it soud like imgir umbers do t eist. The do eist, but the seem strge to us becuse most of us do t use them i d-to-d life, so we hve hrd time visulizig wht the me. However, imgir umbers re etremel useful (especill i electricl egieerig) d mke m of the techologies we use tod (rdio, electricl circuits) possible. The umber i: i is the umber whose squre is. Tht is, i = = d i. We defie the squre root of egtive umber s follows: = = = i or i. Emples: Epress i terms of i. ) 64 b) c) 49 d) 8 Imgir Number: A umber tht c be writte i the form + bi, where d b re rel umbers d b 0. A umber with i i it is imgir. 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 imgir umbers.

14 Addig or Subtrctig Comple Numbers i cts like other vrible i dditio d subtrctio problems. Distribute egtive sigs d combie like terms (dd or subtrct the rel prts d dd or subtrct the imgir prts). Write our swer with the rel prt first, the the imgir prt. Emples: Add or subtrct d simplif. ) ( + i) + ( i) b) ( 4 i) ( + i) c) ( 7i) ( 6) d) i ( i) Multiplig Comple Numbers Multiplig Comple Numbers: To multipl imgir umbers, first write squre roots of egtive umbers i terms of i. Multipl s usul b distributig, FOILig, d usig epoet rules. Tret i like other vrible. Use the fct tht i =. Awhere ou see i, chge it to. o 8i = 8( ) = 8 i = = o ( )( ) Emples: Multipl d simplif. If the swer is imgir, write it i the form + bi. ) 9 4 b) c) i 7i d) ii e) i( i) f) ( 7+ i)( 9 8i) g) ( i) h) ( 4i)( + 4i) Simplif Power of i : Epress the give power of i i terms of powers of i, d use the fct tht Emples: Simplif ech epressio. ) i b) i c) i 7 d) i 47 i =.

15 .8 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) i c) i Dividig Comple Numbers: Multipl both the umertor d the deomitor b the comple cojugte of the deomitor. Emples: Divide d simplif to the form + bi. ) 7 i b) + 6 i i c) 9i 7+ 6i d) + i 4 i

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