Polynomials - Polynomial notes PDF

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POLYNOMIALS*

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This%package%includes%AWESOME%topics%such%as:

• Like%Terms% • Monomials%and%Binomials% • Adding%Polynomials% • Subtracting%Polynomials% • Multiplying%Polynomials% • Multiplying%2%Binomials%using%FOIL% • Dividing%Polynomials% • Finding%the%Greatest%Common%Factor% ! !

EXCITED*YET???* ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

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WHAT%IS%A%MONOMIAL?%

2a2!

COEFFICIENT!=!N

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VARIABLE!=!LETTER!WITH! THE!POWER!TO!BECOME! ANY!NUMBER!IT!WANTS! DEPENDING!ON!THE! SITUATION…SO!COOL!!

Proof:!!!!!!!!2a!=!10!(a=5)! !!!!!!!!!!!!!!!!!!!!!!!!!2a!=!16!(a=8)…told!you!so.!

! Still!a!MONOMIAL….but! there’s!2!variables…that’s! crazy!!!

2a2b3! !

So#What’s#a#POLYNOMIAL???% Polynomials!are!monomials!that!are!linked!by!addition!or! subtraction.! ! Binomial:!a!polynomial!with!two!terms! E.g.!(2a!+!3)!and!(5x2!+!3x)!are!examples…! ! Trinomial:!a!polynomial!with!three!terms! E.g.!(2a2!+!3a!+!5)!and!(5x2!+!3y!+!x)!are!examples…! ! ! ! ! ! ! ! !

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ADDITION'OF'POLYNOMIALS% •

Only!like*terms*can!be!added!or!simplified!to!a!single!term!

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Terms*that*are*not*like*terms*cannot*be*simplified*to*a*single*term.!

! ! !

OK…So*what*is*a*“like*term”?* ! • Two!terms!are!said!to!be!like*terms!when!identical!exponents!are! assigned!to!identical!variables.! !

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Like*terms?*

a)*2x*and*3x*

YES*

b)*M5*and*7**

Yes*

c)*5b3*and*4b2*

No*

d)*x3*and*xy2*

No*

e)*3xy2*and*0.5xy2*

Yes*

f)*2xy2*and*6x2y* *

No*

EXAMPLES:!! ! a) 2x!+!3x!=!5x! b) 5x2!+!6x!=!can’t!be!added!because!the!variables!are!identical!but!the!exponent!is!not! c) 10x2y!+!5x2y!=!15x2y! ! TRY!THESE:!! ! 1. 2x2!+!3x2!=!_________________________! 2. 5ab3!+!10ab3!=!____________________! 3. a3xy2!+!5xy2!=!_____________________! !

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ADDITION'OF'POLYNOMIALS'WITH'FRACTIONS% !

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SECONDARY!1!REVIEW:!!! + ! =!?!!!(Forget!how!to!do!this?)! ! ! ! ! ! ! ! ! !

ex!1 ! ! ! ! ! !

ex!2 ! ! ! ! ! ! ! ! ! ! ! ! !

Step!1:!Find!a!common!denominator! Step!2:!Add!the!numerators!(that’s!the!top!#)! Step!3:!Keep!the!denominator!(you!guessed!it…bottom!#)! Step!4:!Reduce!it!if!you!can! 1 3 1 2 3 5 + ! !!! = !!!! + ! !!! = !!!! !!!"#$%"!!"!!1 ! 4 4 4 4 2 4 ! This should look familiar to all of you

x 3x +x= 2 2

SHOW!WORK!HERE!

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t t t 3t 4t 2t + = + = = 6 2 6 6 6 3

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Adding&fractions&with&variables* Step!1:#Gather#like#terms Step%2:%If%necessary,%find%a%common%denominator Step%3:%Use%equivalent%fractions%to%place%like%terms%on%!! !!!!!!!!!!!!!!a"common"denominator! Step%4:%Add%coefficients%of%like%terms

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ADDITION'PRACTICE% ! PRACTICE!TIME:!! ! 1) (3x!+!2)!+!(2x!+!5)! !

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2)!(4a2!+!3a!+!2)!+!(2a2!+!2a!+!3)!

3) (5x!+!6)!+!(2x!–!4)!

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4)!(2x2!+!3x!+!4)!+!(5x2!a!3x!a!2)!!

5)!(5d2!–!8d!+!4)!+!(d2!+!3d!+!2)!! ! ! ! ! ! ! 7)!(8x2!–!6x)!+!(ax2!+!8x!+!7)! ! ! ! ! ! ! ! ! ! 2a 2 3a a2 a 1 9)!! !( + ! + )!+ ( !+ +! 3 5 4 3 5 ! ! ! ! ! ! ! !

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6)!(a3x2!+!6x)!+!(a2x2!+!8x!–!12)!

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8)!(!

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3 ) ! 4

a 4 1 + ) + (a + ) 2 5 5

10)!(0.5x2!+!3)!+!(a1x2!–!6)!

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TRY$THIS!!!% If#…!!

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A"="2x !+"3x"+"2 2

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! !!!!!!!!B"="a3x !–!2x#–!4

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! !!!!!!!!C"="5x !–!1"

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Find%.%.%. 1) A"+"B! A"+"C! 2) 3) B"+"C!

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MORE%PRACTICE% 1. What!is!the!simplified!algebraic!expression!that!represents!the!perimeter!of!this!! polygon?!

(2x + 5) cm (x – 4) cm

(8 – x) cm (3x – 2) cm

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! ! ! ! ! ! ! ! ! 2. The!perimeter!of!this!rectangle!is!36!cm.!What!is!the!area!of!the!rectangle?!! !!!!!!Give!a!numerical!value!(not!an!algebraic!expression).! !

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(2x – 4) cm (x + 7) cm

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Subtracting+Polynomials% ANOTHER!SECONDARY!1!REMINDER:!A!subtraction!is!just!an!addition!of!an!opposite!! ! ! ! ! ! Examples:!! 5!–!3!=!5!+!(a3)!!=!2! ! ! ! ! ! ! ! a5!–!4!=!a5!+!(a4)!=!a9! ! ! ! ! ! ! ! 2x!–!5x!=!2x!+!(a5x)!=!a3x! ! ! STEPS*FOR*SUBTRACTING*POLYNOMIALS* ! • Step!1:!!Drop!the!brackets! • Step!2:!For!the!bracket!with!the!minus!sign!in!front!of!it,!add!the!opposite!of!each!term! • Step!3:!Gather!like!terms!and!simplify! ! ! ex!1:!!(4x!–!3)!–!(2!+!5x)! ! ! ! Step!1:!4x!–!3!!–!!(2!+!5x)! ! (change!the!minus!to!+)! ! ! ! ! ! ! ! Step!2:!4x!–!3!!+!!(a2)!–!5x! ! ! ! ! ! ! ! ! Step!3:!4x!–!5x!–!3!+!(a2)! ! ! ! ! ! ! ! ! ANS:!!ax!–!5! ! ! ! ! ex2:!(3m2!–!4m!+!3)!–!(2m2!–!4m!+!5)!+!(m2!–!2m)!!! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

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! ex3:!What!is!the!difference!in!the!perimeters?!!! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

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a+1

3 !

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MULTIPLYING*POLYNOMIALS% MULTIPLYING*A*MONOMIAL*WITH*A*BINOMIAL* * * 1. 3(2c!+!2)! ! 2.!!a(2a!–!1)!! 3.!!2(3x!–!2)! 4.!!a4!(aa!–!3)! ! ! ! 6c!+!6!! ! !!!!!!2a2!–!a! ! !!!!!!!6x!–!4! ! !!!!!!!!!4a!+!12! ! ! ! A!TRICK!TO!CHECK!YOUR!WORK! ! ! ! Ex:!3!(x!+!2)! !!!!!!!How!to!check!if!you’re!right…!!!! ! !!!!!!!!!3x!+6! ! ! ! ! ! Original!Question! ! Your!Answer! ! ! ! ! !!!!!3!(x!+!2)! ! ! !!!!!3x!+!6! ! ! ! ! ! ! ! ! ! ! ! ! ! Now$plug$in$“2”$for$“x”$in$both$$ ! ! ! ! ! ! !! ! ! ! ! ! ! !!!!!!3(2!+2)! ! ! !!!!!3(2)!+6! ! ! ! ! ! !!!!!!!!3(4)! ! ! !!!!!!!6!+!6! ! ! ! ! ! ! !!!!!!!!!!12 2! ! ! ! ! ! ! !!!!!!!!!!!!!!!!!(same!answer!=!you’re!right)! ! ! ! ! PRACTICE:!Simplify!the!following.!Then,!check!by!replacing!x!with!2.! ! ! 1. 2(4x!–!8)! ! ! ! ! ! 3.!!!a2x!+!x(x!–!2)! ! ! ! ! ! ! 2. 5a(a+3)!–!(3a2!+!10a)! ! ! ! ! !

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MULITIPLYING*PRACTICE% ! 1) Find!the!algebraic!expression!for!the!area!of!this!rectangle.! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

3x!cm!

(x!–!2)!cm!

2)!If!the!area!of!this!rectangle!is!66!cm2,!what!is!its!perimeter?!(Numerical!value!)! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

3!cm!

(x!–!2)!cm!

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MULTIPLYING*TWO*BINOMIALS% * ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

FOIL* 1. Multiply!the!two!First!terms!of!each!polynomial! 2. Multiply!the!two!Outer!terms! 3. Multiply!the!two!Inner!terms! 4. Multiply!the!two!Last!terms! 5. Simplify! ! (x!+!3)!(x!+!5)! F:!!x2!! O:!5x! I:!3x! L:!15! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!So!your!answer!is!:!!!x2!+!5x!+!3x!+!15! ! !!!But!don’t!forget!to!add!your!like!terms:!!!!!!!x2!+!8x!+!15! ! ! !

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PRACTICE(MAKES(PERFECT!% 1. (2n!+!3)(n!+!2)!

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2.!(3y!+!4)(y!–!4)!

3. !x(2x!+!3)(2x!–!1)!

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4.!!3b(2b!–!5)(¼b!+!2)!

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!!!!!!!!!!!!!!6.!!!!x(x!+!2)!+!2(x!–!4)!

! ! ! ! ! ! ! ! ! ! ! ! ! 5.!!(x!+!2)(x!–!4)! ! ! ! ! ! ! ! ! ! 7. 4s(s2+s)!–!4s2(s!–!1)! ! ! ! ! ! ! ! ! !

! ! ! ! ! 8. What!is!the!area!of!a!rectangle!with!a!length!of!(x!+!2)!and!a!width!of!(2x!–!1)?!

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VERY%IMPORTANT% ! ! !

(a!+!b)2!=!(a!+!b)(a!+!b)!

! WHAT*80%*OF*STUDENTS*DO*ON*JUNE*EXAM* WHAT*YOU*SHOULD*DO* ! * * ! (2x!+!5)2! (2x!+!5)2! ! 4x2!+!25! (2x!+!5)(2x!+!5)! ! ! ! 4x2!+!10x!+!10x!+!25! Please$don’t$let$this$happen!!!$ ! 4x2!+!20x!+!25! ! ! ! ! !!!!!!!!!!!!!!!!!!!!!!BECAUSE……!(2!+!3)2!=!52!=!25!…..NOT…..!22!+!32!=!4!+!9!=!13…!SEE!THE!DIFFERENCE???! ! ! EXAMPLES:! ! ! ! a)!(x!+!1)2! ! ! ! ! ! ! b)!(3a!–!5)2! ! ! ! ! ! ! ! ! ! ! ! ! ! ! c)!!(x!+!4)2! ! ! ! ! ! ! d)!(4a!+!2)2! ! ! ! ! ! ! ! ! ! ! ! ! !

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DIVIDING&POLYNOMIALS% To!divide!a!polynomial!by!a!monomial,!divide!each!term!of!the!polynomial! ! !

! ! ! ! ! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! ! ! ! ! 1. Turn!the!expression!into!a!fraction! ! ! ! ! 2.!Separate!into!several!fractions! ! ! ! 3.!Simplify! ! ! !

(12a + 4) ÷ 4 !

12 a + 4 4

12a 4 + 4 4

3a + 1

PRACTICE(QUESTIONS:( !

1. (3z!–!9)÷3!

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2. (24a!–!12)!÷!a12!!

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3. (ax!–!8)!÷!a1!

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4. (10m2+15m)!÷!5m!

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5. (2x2!+!x3)!÷!x2!

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6. (12n4!–!9n3)!÷!3n2!

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FACTORING*POLYNOMIALS*(Finding*the*GCF)%

! 6a2!+!15a! ! !

1. Determine!the!greatest*common*fa terms!in!the!expression! ! ! ! !

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3a!

2. Divide!the!algebraic!expression!by!t factor! ! ! ! ! ! !

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6a 2 +15a 6a 2 15 a ! ! != + 3a 3a 3a

3. Write!the!product!of!the!factor!from from!(2)! ! ! * ! !

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****3a*(2a*+*5)

4. Check!!

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!!!!!!3a!(2a!+!5)!=!6a2

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PRACTICE!:!! ! 1.!Find!the!GCF!(Greatest!Common!Factor)! ! a) 5x2y!,!25xy!,!15!xy2!:_______________! b) 32h2!,!16h4!,!20h3!:________________! c) ab4!,!35ab!,!7a3b!:__________________! ! 2.!Factor!each!of!these!polynomials! a)!5x2!–!25x!

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b)!6b2!+!12b!

c)!8b2!+!4b!–!4! !

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d)!15rs2!+!12rs!–!9s!

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Word%Problem% The!area!of!a!rectangle!is!(3x3!+!6x2)!cm2.!The!width!of!the!rectangle!is!the! greatest!common!factor!of!the!area.!Find!the!algebraic!expression!for!the! perimeter!of!the!rectangle.!! ! ! ! ! ! ! ! Workbook:! Multiplication/Division!of!monomials:!pg.!51a53! Sum/difference!of!polynomials:!pg.!57a58! Product!of!polynomials:!pg.!59a63! Division!of!polynomials:!pg.!65a66! GCF:!pg.!66a67! !

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