Expanding and Simplifying PolynomialsFactoring – The Decision Tree The Different Types of...
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MCR3U – Unit 1: Rational Expressions – Lesson 1 Date:___________ Learning goal: I can add, subtract, and multiply polynomials. Expanding and Simplifying Polynomials COLLECTING LIKE TERMS Recall: only like terms can be collected through addition and subtraction. a) 5 − 3 + 6 − 2 + 3 b) 4 − 3 + 7 − 3 − 6 + 7 c) 3 ! + 5 ! + 6 ! − 5 ! MULTIPLYING MONOMIALS Recall: multiple coefficients with coefficients and variables with variables. a)(−2)(4) b) (2 ! )(−3 ! ) c) 3 ! (2)(5 ! ) DISTRIBUTION Recall: multiple the term in front of the bracket to each term inside the bracket. a) 4( + 3) b) 4(2 ! − 5 + 7) c) 2 − 4 − 5 − 7 − ( − 2) d 3 ! 2 ! − 4 + 7 − 7(3 ! + 4 ! − 7) MULTIPLYING BINOMIALS Recall: multiply each term in the first bracket by each term in the second bracket (FOIL rule). a) + 3 ( − 4) b) − 5 (3 − ) c) 2 − 1 (3 − 5) d) 5 − 2 !
Transcript of Expanding and Simplifying PolynomialsFactoring – The Decision Tree The Different Types of...
MCR3U–Unit1:RationalExpressions–Lesson1 Date:___________Learninggoal:Icanadd,subtract,andmultiplypolynomials.
ExpandingandSimplifyingPolynomials COLLECTINGLIKETERMS
Recall:onlyliketermscanbecollectedthroughadditionandsubtraction.
a) 5𝑥 − 3𝑦 + 6 − 2𝑦 + 3𝑥 b) 4 − 3𝑥 + 7 − 3𝑦 − 6𝑦 + 7𝑥 c)3𝑎!𝑏 + 5𝑎𝑏! + 6𝑎𝑏! − 5𝑎𝑏!
MULTIPLYINGMONOMIALS
Recall:multiplecoefficientswithcoefficientsandvariableswithvariables.
a)(−2𝑥)(4𝑥) b)(2𝑏!)(−3𝑏!) c)3𝑎!(2𝑎)(5𝑎!)
DISTRIBUTION
Recall:multipletheterminfrontofthebrackettoeachterminsidethebracket.
a)4(𝑥 + 3) b)4𝑥(2𝑥! − 5𝑥 + 7)
c)2 𝑥 − 4 − 5 𝑥 − 7 − (𝑥 − 2) d3𝑏! 2𝑏! − 4𝑏 + 7 − 7𝑏(3𝑏! + 4𝑏! − 7𝑏)
MULTIPLYINGBINOMIALS
Recall:multiplyeachterminthefirstbracketbyeachterminthesecondbracket(FOILrule).
a) 𝑥 + 3 (𝑥 − 4) b) 𝑥 − 5 (3 − 𝑥) c) 2𝑥 − 1 (3𝑥 − 5) d) 5𝑥 − 2!
Simplify 2𝑥 + 1)(3𝑥 − 4)(𝑥 − 2 intwodifferentways.
MethodOne
2𝑥 + 1)(3𝑥 − 4)(𝑥 − 2
MethodTwo
2𝑥 + 1)(3𝑥 − 4)(𝑥 − 2
CONSOLIDATION
Simplifyingeachofthefollowing.
a)(3𝑥 − 2)(4𝑥 − 1)2 b)(2𝑥 + 5)𝟐 − 2 3𝑥 + 7 + (𝑥 − 2)𝟐
c)4 − 2 𝑥 − 3 2𝑥 − 1 d)5 − 2 𝑥 − 3 ! + 2 𝑥 + 4 𝑥 − 1 − 7
HW:ExpandandSimplifyWorksheet
ExpandandSimplifyWorksheetPleasecompleteallworkonaseparatepage.1. 2. 3.
4. 5. 6.
7. 8. 9.
10. 11.
12. 13.
14. 15.
Answers
1. 2. 3.
4. 5. 6.
7. 8. 9.
10. 11.
12. 13.
14. 15.
( )( )33 −+ xx ( )( )82 −+ xx ( )( )105 ++ xx
( )( )132 −+ xx ( )( )832 −+ xx ( )( )2312 ++ xx
( )( )323 −+ xx ( ) ( )1215 2 −+ xx ( )( ) 6105 +++ xx
( )( )21252 +− xx ( )( )( )14323 −−+ xxx
( ) ( )2522523 ++− xx 3x + 2( )2 −3 x + 4( )+ x − 4( )2
( )( ) ( )7531253 +−−+ xxx ( ) ( ) ( )22 1523213 +−+− xxx
( )( )33 −+ xx ( )( )82 −+ xx ( )( )105 ++ xx92 −= x 1662 −−= xx 50152 ++= xx
( )( )132 −+ xx ( )( )832 −+ xx ( )( )2312 ++ xx32 2 −+= xx 1623 2 −−= xx 276 2 ++= xx
( )( )323 −+ xx ( ) ( )1215 2 −+ xx ( )( ) 6105 +++ xx932 2 −+= xx
( )( )185501211025
23
2
−−−=
−++=
xxxxxx
56152 ++= xx
( )( )21252 +− xx ( )( )( )14323 −−+ xxx( )( )51812814452
23
2
−−−=
++−=
xxxxxx ( )( )
821591869
23
2
+−−=
−−−=
xxxxxx
( ) ( )2522523 ++− xx ( ) ( ) ( )22 44323 −++−+ xxx( )35468
2520421562
2
++=
+++−=
xxxxx
8101681234129
2
22
++=
+−+−−++=
xxxxxxx
( )( ) ( )7531253 +−−+ xxx ( ) ( ) ( )22 1523213 +−+− xxx
26862115576
2
2
−−=
−−−+=
xxxxx
( )( ) ( )
136351822050316151811025232169
23
223
22
+−−=
−−−+−+=
++−++−=
xxxxxxxxxxxxx
MCR3U–Unit1:RationalExpressions–Lesson2 Date:___________Learninggoal:Icanidentifywhentocommonfactorbeforefactoringsimpleandcomplextrinomials.
Factoring COMMONFACTORING
Recall:wearelookingforfactorsthatarecommontoallterms.
a)2𝑥 + 50 b) 12𝑥!𝑦 − 18𝑥!𝑦! + 24𝑥!𝑦! c) 3𝑥 𝑥 − 1 − 2(𝑥 − 1)
Note:whenfactoringyoumustALWAYScommonfactorfirstifpossible.
‘SIMPLE’TRINOMIALS
Asimpletrinomialisoneintheform𝒂𝒙 𝟐 + 𝒃𝒙 + 𝒄,where𝒂 = 𝟏.Ifitisfactorable,asimpletrinomialcanbeexpressedasaproductoftwobinomials.Theconstanttermsofthebinomialsareintegersthatmultiplytogive“c”andaddtogive“b”. a)𝑣! + 2𝑣 − 15 b)−2𝑡! + 26𝑡 − 84 c)−𝑛! + 4𝑛 + 32
DIFFERENCEOFSQUARES
Aspecialcircumstancewithtwoterms,wherethefirsttermandthelasttermarebothsquares,andthetwotermsaresubtracted(𝑏 = 0).
a)𝑥! − 𝑦! b) 25𝑡! − 64 c) 𝑐! − 36𝑑!" d)16𝑥! − 81𝑦!
‘COMPLEX’TRINOMIALS
Acomplextrinomialisoneintheform𝒂𝒙 𝟐 + 𝒃𝒙 + 𝒄,where𝒂 ≠ 𝟏.Wearenolongerlookingfortwonumbersthataddtobandmultiplytoc.
a)2𝑘! + 3𝑘 + 1 b)18𝑥! + 33𝑥 + 12 c)27𝑦! − 6𝑦 − 56
d) 2𝑝! − 𝑝! − 28 e) 6𝑥! − 8𝑥! + 2𝑥! f)21𝑎! + 32𝑎𝑏 − 5𝑏!
Canalltrinomialsbefactored?
Factoring–TheDecisionTreeTheDifferentTypesoffactoring:
1. CommonFactoring2. DifferenceofSquares(∆2–!2)3. “Simple”Trinomial(ax2+bx+c,a=1)4. “Tricky”Trinomial(ax2+bx+c,a≠1)
Strategyforfactoring:
HW:FactoringWorksheet
START
CommonFactoringTakethegreatestcommonfactoroutofeachtermintheexpression.
e.g.3x2–12x
Howmanytermsarethere?
2terms
3terms
DifferenceofSquares∆2 –!2 = (∆ +!)(∆ –!)
e.g.16x2–49
ax2+bx+c
a=1
a≠1
“Simple”TrinomialFind2integersthatmultiplyto“c”and
addto“b”
e.g.x2+3x–10
“Tricky”TrinomialFind2integersthatmultiplyto“(a)(c)”andaddto“b”.Replacethe“b”termwithtwotermswhosecoefficientsarethose2
integers,thenfactorbygrouping.
e.g.2x2–x–6
Canwefactoranyofthefactorsagain?If“yes”,RESTART!
ax2-c
FactoringWorksheetFactoreachofthefollowingonaseparatepieceofpaper.1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. b2–36 20. m2–25 21. 81n2–12122. x2–4x+4 23. 9a2+48a+64 24. 32c2–48c+1825. 26. 27. 28. 29. 30. 31. x2+3xy−28y2 32. 25x4−121y2 33. x6−8x3−20 34. 4x4−4x2−15 35. 6x4−8x6+2x3 36. 21a2+32ab−5b2 37. x2−y2 38. x4−y4 39. 2x4−2x2 40. x2y+17xy−60y 41. 2x³y–8xy³ 42. 9x2+64y2
Bonus:Factor:x2(a+b)+10x(a+b)+16(a+b) Answers1.5x2(4x3–11) 2.4x(3+2x3) 3.4x3(1–3x2)(1+3x2)4.6x2y(3+5y4)5.–7xy2(x2y–4)6.9x2y(3+y–7x2y2)7.(x–7)(x–8) 8.(x+10)(x+5) 9.(x–6)(x+5) 10.(x+9)(x–4)11.(x–9)(x–1)12.(x–2)(x–3)13.(x+2)(4x–5)14.(x+2)(3x+1) 15.(x+3)(2x–1)16.(a+2)(3a–2)17.((y–1)(4y+3)18.(a+b)(5a–7b)19.(b–6)(b+6)20.(m+5)(m–5) 21.(9n+11)(9n–11)22.(x–2)2 23.(3a+8)2 24.2(4c–3)225.2(x+5)(x+8)26.–3(x2+x–24) 27.4(x–8)(x+2)28.2(x+3)(x+7)29.2(2x+5)230.9(x–3)231.(x+7y)(x−4y)32.(5x2−11y)(5x2+11y)33.(x3−10)(x3+2)34.(2x2−5)(2x2+3)35.2x3(3x−4x3+1)
36.(7a−b)(3a+5b)37.(x+y)(x−y)38. ( )( )( )2 2x y x y x y− + + 39.2x2(x+1)(x−1)
40.y(x+20)(x−3)41.2xy(x–2y)(x+2y)42.Cannotbefactoredbonus.(a+b)(x+8)(x+2)
25 5520 xx − 4812 xx + 73 364 xx −
522 3018 yxyx + 233 287 xyyx +− 34323 63927 yxyxyx −+
56152 +− xx 50152 ++ xx 2 30x x− −
3652 −+ xx 9102 +− xx 2 5 6x x− +
1034 2 −+ xx 23 7 2x x+ + 22 5 3x x+ −
3a2 + 4a− 4 24 3y y− − 2 25 2 7a ab b− −
80262 2 ++ xx 23 3 72x x− − + 64244 2 −− xx
22 20 42x x+ + 28 40 50x x+ + 81549 2 +− xx
MCR3U–Unit1:RationalExpressions–Lesson3 Date:___________Learninggoal:Icanidentifyrestrictionstosimplifyrationalexpressions.
SimplifyingRationalExpressions WARMUP:Canyousimplifythefollowing? a) !"!
!!!!!"
b)!!!!!!!!
c) !!!!!!
d)!!!!!
!!!!
e) (!!!)(!!!)
(!!!) f)!!!!!!!! g)
!!!!!!! f)
!!!!!!"!!!
ArationalExpressionisafractionwherethe__________________and__________________are
__________________.
RESTRICTIONSWheneveryouareworkingwithalgebraicfractions,itisimportanttodetermineanyvaluesthatmustbeexcluded.Whathappensineachofthefollowingexpressionswhen𝑥 = 3issubstituted? a) !!!
!!! b) !!!
!!!!!!!
Wheneveryouusearationalexpression,youmustidentifyanyvaluesthatmustbeexcluded,theseareknown
asrestrictedvalues.Restrictionsareanyvaluesthatmakethe_____________________________.Itisvery
importantthatyoustaterestrictionsfromthefullyfactoredexpression,beforesimplifying.
Example1:Foreachrationalexpression,determinetherestrictions. a) !!
!!! b) !!!!
(!!!)(!!!!) c) !!
!!!! d) !!!!
!!!!!!"
SIMPLIFYINGRATIONALEXPRESSIONS
SuccessCriteriaforsimplifyingrationalexpressions
1.
2.
3.
Example2:Simplifyeachrationalexpression.Statetherestrictions.
a)!"!!!!"!!!"
b) !!!!!!!"
!!!!"!!!"
c)!!!!!!
!!!!!! d)
!!!!!"!!"!!
!!!!!!"!!"!!
MULTIPLESOFNEGATIVEONEWhenyouarefactoringrationalexpressionsyoumustbecarefuloffactorsthatareoffby-1.
ie.!!!!!!
Example3:Simplifyandstateallrestrictions.
a)!!!!!!!"!"!!!
b) !!!!
!!!!!!!!
APPLICATONS
Example4:Anexpressionforcalculatingthedrugdose(inmg)foryoungchildrenis!""!!!!""!!!!""!!"!!!!"!
,where 𝑡istheirageinyears.
a) Determinethedoseforatwo-year-oldchildbeforesimplifyingb) Determinethedoseforatwo-year-oldchildaftersimplifying c) Determinethedoesforathree-year-oldchild
Example5:Writeanexpressionthatsatisfiesthegivenconditionsineachcase
a) Therestrictionsare-2and5.
b) Therestrictionsare1and-3andtheexpressionis !!!!
insimplestform.
c) Equivalentto!!!!!!
witharestrictionof±4.
d) Equivalentto!!!!!
withadenominatorof12𝑥
e) !!!!!!"!!!!"!!
= !?
HW:Pg353-355#1,3,4,8–14(Ans.Corr9a:− 𝟗𝒙
𝟒𝒚 ,9d:𝒙 ≠ ±𝟓𝒚)
MCR3U–Unit1:RationalExpressions–Lesson4 Date:___________Learninggoal:Icanmultiplyanddividerationalexpressionswhilestateallrestrictions.
MultiplyingandDividingRationalExpressions WARMUP:Simplifyeachofthefollowing.Staterestrictionswherenecessary.
a)!!× !"!" b)!!
!
!!× !!!
!!" c)(!"
!)÷ (!"
!")
MULTIPLYINGRATIONALEXPRESSIONS
Findingcommonfactorsandsimplifyingisfasterthanmultiplyingnumeratorsanddenominators.
Example1:Multiplyeachrationalexpression.Stateanswerinsimplestform.Identifyandstateallrestrictions a) !!"!!!
!!!!"!!!"× !!!!
!!!!!!!" b)!
!!!"!!!"!!!!!!!"
× !!!!"!!!!!
DIVIDINGRATIONALEXPRESSIONSYoumustbecarefulwhenyoudividerationalexpressionsbecauseyoumustconsiderrestrictionsbeforeandafterreciprocatingthesecondterm.
ie.!!÷ !
!doesnotexistbecausewecannotdivide!
! byzero.
Also,!!÷ !
!doesnotexistbecausewecannotdivide3byzero.
SuccessCriteriaformultiplyingrationalexpressions
1.
2.
3.
4.
Example2:Divideeachrationalexpression.Stateanswerinsimplestform.Identifyandstateallrestrictions
a) !!!!÷ !!!!!!!"
!! b) !
!!!"!!
÷ !!!!"!!
c) !"!!!!
!!!!!
d) !!!!!"!!!!!!!!!"!
÷ !!!!!!!!
!!!!!!× !!!!!
!!"
HW:Pg359-360#(4-8)ace,17ab
SuccessCriteriafordividingrationalexpressions
1.
2.
3.
4.
5.
MCR3U–Unit1:RationalExpressions–Lesson5 Date:___________Learninggoal:Icanfindacommondenominatorandadd/subtractrationalexpressions.
AddingandSubtractingRationalExpressions WARMUP:Evaluateeachofthefollowing.
a)!!+ !
! b)!!
!+ !!
!" c)!
!− !
!d)!!
!
!"− !!!
!
Example1:Simplifyeachofthefollowing.Stateanyrestrictions.
a) !
!− !
!!− !
!! b) !
!!!!"+ !
!!!!"
SuccessCriteriaforaddingandsubtractingrationalexpressions
1.
2.
3.
4.
5.
c) !!!!!!
!!!!!!!− !"!
!!!!" d) !
!!!!!!!− !
!!!!!!!
e) !!!!!!
+ !!!!!!
f)!!!!!!!!
− !!!!!!!!!!
!!!!!!÷ !!!!!"!!!
!!!!"
HW:Pg.370-371#5ace,6ace,7,9aef,19
MCR3U–Unit1:RationalExpressions–Lesson6 Date:___________Learninggoal:Icanfindacommondenominateandevaluaterationalequations.
SolvingEquationswithRationalExpressions WARMUP:Solveeachofthefollowingusingoppositeoperations.a)3𝑥 − 7 = 14 b) 3𝑥! − 2𝑥 + 5 = 2𝑥! − 3𝑥 + 25 Recall:Wecansolveequationswithfractionsbyusingacommondenominator.Unlikesimplifyingexpressions,whensolvinganequationwehavetheluxuryofbeingabletomultiply(aslongaswemultiplyonbothsides).
Wedon’tneedtomakethedenominatorscommon,sincewecanclearallfractionsbymultiplyingbyacommondenominator.
Example1:Evaluatebyclearingfractions.
a)!!− !!!
!= 4+ !!!!
!
SuccessCriteriaforsolvingrationalexpressions
1.
2.
3.
4.
Example3:Solveeachrationalexpression.
a) !!!!
+ !!!!!!!!!
= !!!!
b) !!!!
+ !!!!!
= 5c) !
!!!!!!!"= !
!!!+ !
!!!
HW:Pg.370#11,puzzle132
