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+ −