collecting like terms - engage explore...
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MPM1D–Unit2:Algebra–Lesson5 Date:______________Learninggoal:howtosimplifyalgebraicexpressionsbycollectingliketerms.
CollectingLikeTerms
WARM-UPExample1:Simplifyeachexpressionusingexponentlaws.a) (3!!!)(−7!!)
b) (4!!!!!)(6!!!) c) !!!(!!!) !!!!!
Recall,liketermsaretwoormoretermsthathavethesamevariableraisedtothesameexponent.
Algebraicexpressionsthatcontainliketermscanbesimplifiedbycombiningeachgroupofliketermsintoasingleterm.Examples: 3x+4x 9x2–6x2 12x3y2-5x3y2Whycan’tyousimplify? 4x2+4x x2–7 6x3y+5xy3 Example2:Simplifythefollowingalgebraicexpressions.a) 7! + 5 – 3! b) 6!! + 11! + 8! !– 15!
c) 6! + 4 – 5+ 7!
d) −12! – 5 – 7! – 11 e) 2! ! − 3! + 7 − 3!! + 4! – 7
f) 11!!! – 12!!!
Assignment2.5:CollectingLikeTerms1. Arethetermsineachpairlikeorunlike?
a) 5aand–2a b)3x2andx3 c)2p3and–p3 d)4aband ab
e) –3b4and–4b3 f) 6a2band3a2b g)9pq3and–p3q h)2x2yand3x2y22. Simplify.
a) 4+v+5v–10 b)7a–2b–a–3b c)8k+1+3k–5k+4+k
d) 2x2–4x+8x2+5x e)12–4m2–8–m2+2m2 f) –6y+4y+10–2y–6–y3. Simplify.
a) 2a+6b–2+b–4+a b)4x+3xy+y+5x–2xy–3y
c) m4–m2+1+3–2m2+m4 d)x2+3xy+2y2–x2+2xy–y24. Findtheperimeterofeachfigurebelow.a) b) 2.5Answers1. a) like b) unlike c) like d) like e)unlikef ) like g)unlikeh)unlike2. a) 6v−6b) 6a−5b c) 7k+5 d) 10x2+x e)4−3m2 f ) −5y+43. a) 3a+7b−6 b) 9x+xy−2y c) 2m4−3m2+4 d)5xy+y24.a)8n+4 b)8y+2
32
n
MPM1D–Unit2:Algebra–Lesson6 Date:______________Learninggoal:howtoaddandsubtractpolynomials.
AddingandSubtractingPolynomials
WARM-UPExample1:Simplifyeachalgebraicexpression.a) 6! + 4 – 5+ 7!
b) 3! + 7 − 3!! + 4! – 7 c) 11!!! – 12!!!
Example2:Determineasimplifiedexpressiontheperimeterofthefollowingrectangle.ADDINGPOLYNOMIALSInordertodistinguishonepolynomialfromanotherpolynomialinanalgebraicexpression,thepolynomialsareoftenplacedinseparatepairsofbrackets. ie.(3!! + 5! − 1) + (4!! − 2!)Example3:Addthefollowingpolynomials.a) (4! − 7) + (−3! + 2) b) !! + 2! + 6!! + 10! + (5! + 1)
SUBTRACTINGPOLYNOMIALSSupposeyouwereaskedtoevaluatethefollowingexpression:4− (−2).Explaintheprocedureforsubtractingintegers.Thissameideacanbeusedtosubtractpolynomials.Example4:Subtractthefollowingpolynomials.a) 3! − 2 − (4! + 9)
b) 5!! − 12! − (−4! − 8)
c) 12− 4!! + 5! − 1 d) 8!! + 7! − 4!! − 3 − (−5! + 9)
Example5:Themeasuresoftwosidesofatrianglearegiven.TheperimeterisP=4x2+5x+5.Findthemeasureofthethirdside.
x2+3x–5
2x2+3x+6
Assignment2.6:AddingandSubtractingPolynomials1. Simplifythefollowingexpressions.
a)(y2+6y–5)+(–7y2+2y–2) b) (–2n+2n2+2)+(–1–7n2+n)c) (3m2+m)+(–10m2–m–2) d)(–3d2+2)+(–2–7d2+d)e)(4–8w)–(7w+1) f)(mn–5m–7)–(–6n+2m+1)g)(xy–x–5y+4y2)–(6y2+9y–xy) h)(2a+3b–3a2+b2)–(–a2+8b2+3a–b)
2. Foreachshapebelow,writetheperimeterasasumofpolynomialsandinsimplestform. i) ii) iii) iv)
3. Astudentsubtracted
(3y2+5y+2)–(4y2+3y+2)likethis:=3y2–5y–2–4y2–3y–2=3y2–4y2–5y–3y–2–2=–y2–8y–4a) Explainwhythestudent’ssolutionisincorrect.b) Whatisthecorrectanswer?Showyourwork.
4. Thedifferencebetweentwopolynomialsis(5x+3).Oneofthetwopolynomialsis(4x+1–3x2).Whatistheotherpolynomial?Explainhowyoufoundyouranswer.
5. Thesumoftheperimetersoftwoshapesisrepresentedby13x+4y.Theperimeterofoneshapeisrepresentedby4x–2y.Determineanexpressionfortheperimeteroftheothershape.Showyourwork.
6. Arectangularfieldhasaperimeterof10a–6meters.Thewidthis2ameters.Determineanexpressionforthelengthofthisfield.
2.6Answers1. a) –6y2+8y–7 b)–n–5n2+1 c)–7m2–2 d)–10d2+d e) 3–15w f)mn–7m–8+6n g)2xy–x–14y–2y2 h)–a+4b–2a2–7b2
2. i)(2n+2)+(n+1)+(2n+2)+(n+1)=6n+6 ii)(3p+4)+(3p+4)+(3p+4)=9p+12 iii)(4y+1)+(4y+1)+(4y+1)+(4y+1)=16y+4 iv)(a+8)+(a+3)+(12)=2a+233. a) Thestudentisincorrectbecausehechangedthesignsinthefirstpolynomial.
b) (3y2+5y+2)–(4y2+3y+2)=3y2+5y+2–4y2–3y–2=3y2–4y2+5y–3y+2–2=–y2+2y4. (4x+1–3x2)–(5x+3)=–3x2–x–2,or(5x+3)+(4x+1–3x2)=–3x2+9x+45. 9x+6y6. 3a-3
MPM1D–Unit2:Algebra–Lesson7 Date:______________Learninggoal:howtomultiplyapolynomialbyamonomialtosimplifyexpressions.
MultiplyingaPolynomialbyaMonomial
WARM-UP
Example1:Simplifythefollowingalgebraicexpressions.
THEDISTRIBUTIVEPROPERTY
Arectanglehasanunknownlengthandawidthof4units.Ifthelengthisincreasedby7unitstocreatealargerrectangle,writeasimplifiedalgebraicexpressionfortheareaofthenew,largerrectangle. Thispropertyisknownasthedistributiveproperty.Thisisalsoknownasexpanding.
a) 6! − 4 + (2! + 4)
b) 2!! +! + 12 − (3!! + 4! − 6) c) (−4!!!)(3!!!!)
d) (4!!)(2!!) e) ! − 6 − 2 − 5! + (! + 4)
f) (!!!!!)(!!!!)(!!!)!
Whatisthewidthofthenewrectangle?Whatisthelengthofthenewrectangle?Whatistheareaofthenewrectangle?
4
xoriginal
7
7
4
x
DistributiveProperty:! (! + !) = !"+ !"
Example2:Expandthefollowing.a) 3(! + 4) b) −7(! + 3) c) −(2! − 1) d) −4(−! − 5)Nowletstryexpandingwithavariable…Simplify: !(2! + 5)Example3:Expandthefollowing.a) !(! + 1) b) 3!(! + 4) c) – !(−5! + 2) d) −3!(2! − 1)Nowletstryexpandingwithacoefficientandavariable…Simplify: 3!(9!! − 4!)Example4:Expandthefollowing.a) 3!(−4!! + 2!!) b) 5!!(3! − 1) c) −2!!(!! − 3! + 9) d) (! − 1)(11!)Nowtrycombiningeverythingyoulearnedaboutsimplifyingexpressions…Simplify: 2 !! − 4! + 3 + 5!(! + 4)
Example5:Expandthefollowing.a) −3 ! − 2 + 6(! + 1)
b) ! !"! − 4! + 3 + 2!(!!! + ! + 4)
c) 3[−2 6− ! + 5!] d) −5! ! + 5 − 2(3!! − 4! − 7)
Assignment2.7:MultiplyingaPolynomialbyaMonomial1. Determineeachproduct.
a) 4(3a+2) b) (d2+2d)(–3) c)2(4c2–2c+3)
d)–4(b2–2b–3) e)5c(c2–6c–1) f)–3h(4–h2)
2. Hereisastudent’ssolutionforamultiplicationquestion.
(–5k2–k–3)(–2)=–2(5k2)–2(k)–2(3)=–10k2–2k–6a) Explainwhythestudent’ssolutionisincorrect.b) Whatisthecorrectanswer?Showyourwork.
3. Writeasimplifiedexpressionfortheareaofthefollowingrectanglesa) b)
4. Expanda)4x2(3x+2) b)2n(2n–3) c)pq(3p+2q)d)3d(2d2–4d+1)e)2(x+4)–4(2x+3)f)3a(2a+4b–3)–2b(3a+2ab) g)2p(p–4)+6(p2+4p–3)
2.7Answers1.a)12a+8b)–3d2–6d c)8c2–4c+6d)−4b2+8b+12e)5c3−30c2−5cf)−12h+3h3 2. a) Thenegativesignswereomittedonthefirstpolynomialwhen(–2)wasdistributed;
(–5k2)(–2)+(–k)(–2)+(–3)(–2)=10k2+2k+63. a) 2d(3d+4)=6d2+8d b)y(4y+6)=4y2+6y4. a)12x3+8x2b)4n2–6nc)3p2q+2pq2d)6d3-12d2+3d e)-6x–4 f)6a2+6ab–9a–4ab2
g)8p2+16p–18
MPM1D–Unit2:Algebra–Lesson8 Date:______________Learninggoal:howtoapplyknowledgeofsimplifyingexpressionstogeometricproblems.
SimplifyingAlgebraicExpressions
WARM-UP
Example1:Simplifyandexpandthefollowing.a) !(! − 5) b) −2!(3! + 1)
c) (−3!!)(3! + 1 − 2!!)
d) 2 3! + 1 + 3(! − 4)
e) 4! 3!! − 2! + 1 − 3!(2!! − 5)
f) 3! 4! − 5! − 2!(2! + 3!)
g) 2! − 3![5 − 2! − 1 ]
h) !! !
! − 3! − !! ! +
!! !
! i) 5!! ! + 6 − 2[! − 2 1 + 2!! ]Example2:Findtheperimeter.
Example4:Findthemissingsidelengthgiventheperimeterbelow.Example5:Findtheareaoftheshadedregion.
Assignment2.8:SimplifyingAlgebraicExpressions
1. Simplifythefollowing:a) 3p–4q+2p+3+5q–21 g)2x2+3x–7x–(-5x2)
b) -3b(5a–3b)+4(-3ab–5b2) h)3x(x-2y)–4(-3x2-2xy)
c) -3(x2+3y)+5(-6y–x2) i)-3(7xy-11y2)–2y(-2x+3y)
d) 1 2 2 43 3 5 7x y x y− − + j)
2 3 4 55 8 15 12s t s t− − −
e) [ ]3 6 2( )x y− + k)2x(x–2y)–[3–2x(x–y)]
f) [ ]{ }3 7 2 (2 1)x x x− − − − l)2 3! − ! ! + ! − 3[! + 4! 3! − 2! ]
2. Atrianglehassidesoflength2acentimeters,7bcentimeters,and5a+3centimeters.Whatistheperimeterofthetriangle?
3. Asquarehasasideoflength9x–2inches.Eachsideisshortenedby3inches.Whatistheperimeterofthenewsmallersquare?
4. Atrianglehassidesoflength4a–5feet,3a+8feet,and9a+2feet.Eachsideisdoubledinlength.Whatistheperimeterofthenewenlargedtriangle?
5. Findtheareaoftheshadedregion.
a) b)
2.8Answers1.a)5p+q-18 b)-27ab-11b2 c)-8x2-39y d)!!!" ! −
!!" ! e)-6x-6y+18 f)-27x+6 g)7x2-4x
h)15x2+2xy i)-17xy+27y2 j) !!" ! −!"!" ! k)4x2-6xy-3 l)3x-38y2+22yw
2.7a+7b+33.36x-204.32a+105.a)116x2+53x b)11x2-44x
MPM1D–Unit2:Algebra–Lesson9 Date:______________Learninggoal:howtofindthegreatestcommonfactortocommonfactorapolynomial.
CommonFactoring
WARM-UP
Example1:Simplifythefollowing.
a) !!"!! !! b) !!!
!!! c) !!"!!!! !!!!! d) !"!!!!
!"!!!
DIVIDINGAPOLYNOMIALBYAMONOMIAL
Rule:whendividingbyamonomial,eachtermmustbedividedbythemonomial.Example2:Simplify.
a) !!!!! !! b) !!!!"!!!"
! c) !"!!!!!"!!!!!"!" !!"
FACTORING
Afactorisanumberortermthatdividesevenlyintoeachtermorpolynomial.Inalgebra,tofactormeanstoexpressapolynomialasaproductoffactors,usuallyamonomialxpolynomial.ThemonomialfactoristheGreatestCommonFactor(GCF)andthepolynomialfactoristheresultofdividingeachtermintheoriginalpolynomialbythemonomialfactor.RecalltheGreatestCommonFactor(GCF)isthehighestvaluethatdividesexactlyintotwoormorevalues.Example3:Givenonefactorofapolynomial,determinetheotherfactorforeachofthefollowing.a) 5isafactorof15!.
b) 3!isafactorof15!".
c) 8!istheGCFof24!" − 16!. d) −!!!istheGCFof−2!!!! + 5!!!-4!!!.
Example4:DeterminetheGCFofeachofthefollowingterms.a) 8!!and16! b) 15!"#,25!",10!!!! c) 7!!!!!!and2!!!!!Factoringistheoppositeofexpanding.
3(10! + 3) 30! + 9Nowletscombineeverythingtofactorabinomial…Consider: Firstlet’sdeterminetheGCF=__________Now,divideeachtermintheoriginalexpressionbytheGCF
Tocompletethisreversedistributiveprocess, =_______()WritetheGCFinfrontofthebrackets,and GCFwhat'sleftafterWhatisleftoverafterdividinginthebrackets. Example5:Commonfactorthefollowing.a) 10! − 20
b) 22! − 99!
c) !! − !
d) −10!! − 25! e) 13! + 12! − 4!
f) 4!! + 8! − 10!"
g) 12!!!!! − 8!"# h) −16!! + 8! + 4! i) 35!!!! − 21!!! + 7!!!!
Factored Form
Expanded Form
FACTORING
EXPANDING
20x2 +15x
Example6:Theareaofatenniscourtisrepresentedby60x2+75x.Whatarethedimensionsofthetenniscourt? Example7:Atrianglehasanareaof andaheightof4x.Whatisthelengthofitsbase?
Theformulafortheareaofatriangleis
28 12x x+
2bhA =
Assignment2.9:CommonFactoring1. Usingthegreatestcommonfactor,writethebinomialinfactoredform.
a)4x+20 b)5x+30x2 c)12x2−48x d)21x2−49xe)−18x+33 f) 20x−50x2 g)−48x2−63x h)−36x3−72x2
2. CommonFactor
a)4x2+12x+8 b) 3x2+6x−9 c)5x3+10x2−120x d)3x4−36x3+105x2
e) f) g) h)
i) j) 3. Theareaofachalkboardisrepresentedby21x2+6x.Whatarethedimensionsofchalkboard?4. Challenge:CommonFactor
a)a2b3c–ab2c2+a2b2c2
b)3x(x+y)+2y(x+y)c)5x(2x–3)–(2x–3)
2.9Answers1. a)4(x+5) b)5x(1+6x) c)12x(x−4) d)7x(3x−7) e)−3(6x+11) f) 10x(2−5x) g)−3x(16x+21) h)−36x2(x+2)2. a)4(x2+3x+2) b)3(x2+2x−3) c)5x(x2+2x−24) d)3x2(x2−12x+35)
e) f) g) h)
i) j) 3.3xby7x+24.a)ab2c(ab–c+ac) b)(x+y)(3x+2y) c)(2x–3)(5x–1)
2 218 50x y− 9 6 5100 50 75z z z+ − 2 2 336 108rs r s− 7 10a b a−5 4 4 32 3 4c d c c− + 3 2c c c+ −
2 22(9 25 )x y− 5 425 (4 2 3)z z z+ − 236 (1 3 )rs rs− 7 3( )a b a−3 2 4(2 3 4)c c d c− + 2( 1)c c c+ −