3A Using pronumerals 3B 3C 3D 3E Contents 3F 3G 3H 3I · 52 maths Quest 9 for the Australian...
Transcript of 3A Using pronumerals 3B 3C 3D 3E Contents 3F 3G 3H 3I · 52 maths Quest 9 for the Australian...
3A Using pronumerals 3B Algebra in worded problems 3C Simplifying algebraic expressions 3D Expanding brackets 3E Expansion patterns 3F More complicated expansions 3G The highest common factor 3H More factorising using the highest
common factor 3I Applications
WhAt Do You knoW?
1 List what you know about algebra. Create the first two columns of a K W L chart to show your list.
2 Share what you know with a partner and then with a small group.
3 As a class, create a large K W L chart that shows your class’s knowledge of algebra.
3
opening Question
Belinda works for an advertising company that produces billboard advertisements.The cost of a billboard is based on the area of the sign and is $50 per square metre. If this billboard has its length increased by 2 m and its height by 3 m, would the increase in cost depend on the initial size of the billboard?
number AnD AlgebrA • pAtterns AnD AlgebrA
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Algebra
ContentsAlgebraAre■you■ready?Using■pronumeralsUsing■pronumeralsAlgebra■in■worded■problemsAlgebra■in■worded■problemsSimplifying■algebraic■expressionsSimplifying■algebraic■expressionsExpanding■bracketsExpanding bracketsExpansion■patternsExpansion■patternsMore■complicated■expansionsMore■complicated■expansionsThe■highest■common■factorThe■highest■common■factorMore■factorising■using■the■highest■common■factorMore■factorising■using■the■highest■common■factorApplicationsApplicationsSummaryChapter■reviewActivities
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50 maths Quest 9 for the Australian Curriculum
Are you ready?Try■the■questions■below.■If■you■have■diffi■culty■with■any■of■them,■extra■help■can■be■obtained■by■completing■the■matching■SkillSHEET■located■on■your■eBookPLUS.
Alternative expressions used for the four operations 1 Write■each■of■the■following■as■a■mathematical■sentence.
a The■sum■of■6■and■4b The■product■of■2■and■5c The■difference■between■3■and■7
Algebraicexpressions 2 Match■the■correct■algebraic■expression■on■the■right■with■each■of■the■descriptions■on■the■left.
a x■is■divided■by■y■ A 3xyb The■sum■of■x■and■y B x - yc 3■times■the■product■of■x■and■y C y■-■3x
d The■difference■between■x■and■y■ D xy
e 3■times■x■is■subtracted■from■y■ E x■+■y
Substitutionintoalgebraicexpressions 3 If■x■=■2■and■y■=■5,■evaluate■each■of■the■following.
a 9x■ b -3yc x■-■y■ d 2x■+■5e 7y■-■10■ f 8xyg 3x2y■ h 6x■-■2y
Liketerms 4 Select■the■like■terms■from■each■of■the■following■lists.
a 3a,■3,■-4a,■2c,■a■ b 5x,■xy,■2y,■x,■12x
c 7qp,■7,■7q,■7pq,■7p■ d ab,■ac,■bc,■a2,■2ac,■c
Collecting like terms 5 Simplify■each■of■the■following■expressions.
a 8y■+■5y■ b 2n■+■4m■+■nc 10x■+■4■-■3x■ d 7k■+■3p■+■2k■-■p
Multiplying algebraic terms 6 Simplify■each■of■the■following.
a 4x■ì■3■ b 3a■ì■7bc -5k■ì■p■ d 2mn■ì■3m
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6■+■42■ì■5
7■-■3
18 -15-3 9
25 8060 2
3a,■-4a,■a 5x,■x,■12x
7qp,■7pq ac, 2ac
13y 3n■+■4m7x■+■4 9k■+■2p
12x 21ab
-5kp 6m2n
2 a■ D b E c A d B e C
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51Chapter 3 Algebra
Dividing algebraic terms 7 Simplify■each■of■the■following.
a 12x■ó■4■ b 15y■ó■yc 8a■ó■2a■ d -21xy■ó■3
Expanding brackets 8 Expand■each■of■the■following.
a 2(x■+■3)■ b 3(y■-■k)c -5(m■+■2)■ d -7(a■-■4)
Finding the highest common factor 9 Find■the■highest■common■factor■for■each■of■the■following.
a 3■and■15■ b 8■and■20c 25■and■35■ d 6a■and■12e 27x■and■36x■ f 5ab■and■10a
Factorising by finding the highest common factor10 Factorise■each■of■the■following■expressions■by■fi■rst■fi■nding■the■highest■common■factor.
a 4m■+■8■ b 2x2■-■6x c 12ab■+■9a
Adding and subtracting fractions11 Calculate■each■of■the■following.
a 16■+2
5■ b 2
3- 5
12 c 7
9+4
5
Multiplying and dividing fractions12 Calculate■the■following,■expressing■the■answer■in■simplest■form.
a 12 ì 2
3■ b 3
4■ì■ 8
27
c 38■ó■2
3■ d 5
21■ó■ 5
14
Simplifying algebraic fractions13 Simplify■each■of■the■following■fractions.
a3 2
6( )x +
■ bx x
x( )− 5
■ c4 72 7
( )( )xx
++
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3x 154 -7xy
2x■+■6 3y■-■3k-5m■-■10 -7a■+■28
3 45 6
9x 5a
4(m■+■2) 2x(x■-■3) 3a(4b■+■3)
1730
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7145
1
3
29
9
1623
x + 2
2x■-■5 2
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using pronumeralsthe language of algebra: use of pronumerals
■■ Algebra■is■a■type■of■language■used■in■mathematics.■■ Pronumerals■(letters■or■groups■of■letters)■are■used■to■represent■unknown■numbers.
■■ Pronumerals■can■also■be■used■to■describe■variables■(varying■values).
■■ Some■important■words■used■in■algebra■are:■equation,■expression,■term,■coeffi■cient■and■pronumeral.
4xy■+■5x■-■3y■=■7xy■+■y■-■2This■can■be■broken■down■as■follows:
■■ Term:■ ■A■group■of■letters■and■numbers■that■form■an■expression■and■are■separated■by■an■addition■or■subtraction■sign
■■ Coeffi cient:■ The■number■part■of■the■term■■ Pronumeral:■ The■letter■part■of■the■term■■ Constant term:■ ■The■term■that■does■not■have■a■pronumeral■attached■to■it.■The■constant■term■is■
independent■of■the■pronumeral■(or■variable).■■ Expression:■ ■A■mathematical■statement■made■up■of■terms,■operation■symbols■and/or■
brackets.■It■does■not■contain■an■equality■sign.■■ Equation:■ ■A■mathematical■statement■containing■a■left-hand■side,■a■right-hand■side■and■
an■equality■sign■between■them■■ Sum:■ ■To■fi■nd■the■sum■of■algebraic■terms■we■add■and■simplify■if■the■terms■have■
like■pronumerals.■■ Difference:■ ■To■fi■nd■the■difference■between■algebraic■terms■we■subtract■and■simplify■if■
the■terms■have■like■pronumerals.■■ Product:■ ■To■fi■nd■the■product■of■algebraic■terms,■the■terms■are■multiplied.
Answer the following for the expression 6x - 3xy + z + 2 + x2z + y2
7.
a State the number of terms.b State the coeffi cient of the second term.c State the coeffi cient of the last term.d State the constant term (if there is one).e State the term with the smallest coeffi cient.f State the coeffi cient of thex term.
3A
Coeffi cientTerm Pronumeral
Term Term Term Constant term
4xy + 5x - 3y = 7xy + y - 2
Expression Expression
Equation
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53Chapter 3 Algebra
think Write
a Count■the■number■of■terms.The■terms■are■6x,■-3xy,■z,■2,■x2z■and■y
2
7.
a There■are■6■terms.
b Identify■the■second■term■(-3xy).■The■number■part■is■the■coeffi■cient.Note:■The■sign■must■accompany■the■coeffi■cient.
b The■coeffi■cient■of■the■second■term■is■-3.
c Identify■the■last■term■ y2
7
.■The■number■part■is■the■
coeffi■cient.c The■coeffi■cient■of■the■last■term■is■1
7.
d Identify■the■constant■term,■that■is,■a■term■with■no■pronumeral.
d The■constant■term■is■+2■or■2.
e Identify■the■smallest■coeffi■cient■and■write■the■whole■term■to■which■it■belongs.
e The■term■with■the■smallest■coeffi■cient■is■-3xy.
f Identify■the■term■that■has■onlyx■in■it■and■write■the■number■that■is■at■the■beginning■of■the■term.■(Note■that■-3xy■is■not■thexterm.■The■coeffi■cient■of■thexyterm■is-3.)
f Thex■term■is■6xso■the■coeffi■cient■is■6.
■■ Pronumerals■are■used■to■write■general■expressions■or■formulas■that■will■allow■us■to■make■a■substitution■for■the■pronumeral■when■the■value■becomes■known.
■■ When■writing■a■general■expression,■we■choose■a■pronumeral■that■can■be■easily■identifi■ed■as■belonging■to■the■unknown■quantity■that■it■represents.■
■■ The■pronumeral■represents■a■number.■■■ It■is■not■a■description■of■the■object.
Write the following sentences using algebra.a A number 6 more than Ben’s ageb The product of a and wc One more than the age difference between Albert and his son Walterd Five times an unknown quantity is added to six times another unknown quantity.
think Write
a 1 Since■Ben’s■age■is■unknown,■use■a■pronumeral. a Let■b■=■Ben’s■age.
2 Six■more■means■‘add■6’. The■number■is■b■+■6.
b ‘Product’■means■multiply. b aw
c 1 Choose■pronumerals■to■represent■Albert’s■age■and■Walter’s■age.
c Let■a■=■Albert’s■age.Let■w■=■Walter’s■age.
2 The■age■difference■between■Albert■and■Walter■is■a■-■w.■Add■1■more■to■this■difference.
a■-■w■+■1
d 1 Choose■pronumerals■for■the■2■unknown■quantities.
d Let■x■=■the■fi■rst■unknown■quantity.■Let■y■=■the■second■unknown■quantity.
2 The■sentence■can■be■broken■into■three■instructions:■5■times■an■unknown■quantity■(5x)■■.■■.■■.■■is■added■to■(+)■■.■■.■■.■■.■■.■■.■■6■times■another■unknown■quantity■(6y).
5x■+■6y
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substitution and formulas■■ In■mathematics,■science■and■engineering,■algebraic■expressions■and■formulas■are■commonly■used.■
■■ For■example,■in■previous■years■you■learned■the■formula■A■=■pr2,■which■enabled■you■to■find■the■area■of■a■circle■with■a■known■radius■of■r.■
■■ We■will■now■look■at■how■to■substitute■particular■values■for■the■pronumerals■in■an■expression■or■formula.
substitution■■ We■can■evaluate■(find■the■value■of■)■an■algebraic■expression■if■we■replace■the■pronumerals■with■their■known■values.■This■process■is■called■substitution.■
■■ Consider■the■expression■4x■+■3y.■If■we■substitute■the■known■values■x■=■2■and■y■=■5,■we■obtain■
■ 4■ì■2■+■3■ì■5■=■8■+■15 =■23
■■ Rather■than■showing■the■multiplication■signs,■it■is■common■in■mathematics■to■write■the■substituted■values■in■brackets.■We■would■write■the■example■above■as:
■ 4x■+■3y■=■4(2)■+■3(5) =■8■+■15 =■23
If x = 3 and y = -2, evaluate the following expressions.a 3x + 2y b 5xy - 3x + 1 c x2 + y2
think Write
a 1 Write■the■expression. a 3x■+■2y
2 Substitute■x■=■3■and■y■=■-2. =■3(3)■+■2(-2)
3 Evaluate. =■9■-■4=■5
b 1 Write■the■expression. b 5xy■-■3x■+■1
2 Substitute■x■=■3■and■y■=■-2. =■5(3)(-2)■-■3(3)■+■1
3 Evaluate. =■-30■-■9■+■1=■-38
c 1 Write■the■expression. c x2■+■y2
2 Substitute■x■=■3■and■y■=■-2. =■(3)2■+■(-2)2
3 Evaluate. =■9■+■4=■13
substitution into formulas■■ A■formula■expresses■one■quantity■in■terms■of■one■or■more■quantities.■■■ Pronumerals■are■used■in■these■formulas■to■represent■the■unknown■quantities.■■ We■know■that■the■formula■for■the■area,■A,■of■a■rectangle■is■given■by:
Area■=■A■=■length■ì■width■=■lw
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55Chapter 3 Algebra
■■ If■a■particular■type■of■rectangular■kitchen■tile■has■length■l■=■20■cm■and■width■w■=■15■cm,■we■can■substitute■these■values■into■the■formula■to■fi■nd■its■area.
A■=■lw =■20■ì■15 =■300■■cm2
■■ If■we■are■given■the■area■of■the■rectangular■tile■to■be■400■■cm2■and■the■width■to■be■55■■cm,■then■we■can■substitute■these■values■into■the■formula■to■calculate■the■length■of■the■rectangular■tile.
A■=■lw■ 400■=■l■ì■55
l■=■40055
■ =■8011
■ ö■7.3■■cm
The formula for the voltage in an electrical circuit can be found using the formula known as Ohm’s Law:
V■=■IRwhere I = current in amperes R = resistance in ohms V = voltage in volts. a Calculate V when: i I = 2 amperes, R = 10 ohms ii I = 20 amperes, R = 10 ohms.b Calculate I when V = 300 volts and R = 600 ohms.
think Write
a i 1 Write■the■formula. a i V■=■IR
2 Substitute■I■=■2■and■R■=■10. =■(2)(10)
3 Evaluate■and■express■the■answer■in■the■correct■units.
=■20■voltsThe■voltage■is■20■volts.
ii 1 Write■the■formula. ii V■=■IR
2 Substitute■I = 20■and■R■=■10. =■(20)(10)
3 Evaluate■and■express■the■answer■in■the■correct■units.
=■200■voltsThe■voltage■is■200■volts.
b 1 Write■the■formula. b V■=■IR
2 Substitute■V■=■300■and■R■=■600. 300■=■I(600)
3 Evaluate■and■express■the■answer■in■the■correct■units.
I■=■300600
■=■12■ampere
The■current■is■12■ampere.
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56 maths Quest 9 for the Australian Curriculum
The distance (x) travelled by an object in a straight line is given by the formula: x = ut + 12at2, where
u is the starting speed in m/s, a is the acceleration in m/s2 and t is the time in seconds.a A car starts at a speed of 50 m/s and accelerates at 6 m/s2 for 7 s. How far has the car travelled?b If the car travels for 500 m in 10 seconds with a constant acceleration of 7 m/s2, what was the initial
speed of the car?
think Write
a 1 Write■the■formula. a x■=■ut■+■12at2
2 Substitute■u■=■50,■a■=■6■and■t■=■7. =■(50)(7)■+■12(6)(7)2
3 Evaluate■and■express■the■answer■in■the■correct■units.
=■350■+■147=■497■■m
The■distance■travelled■is■497■■m.
b 1 Write■the■formula. b x■=■ut■+■12at2
2 Substitute■x = 500,■a■=■7■and■t■=■10. 500■=■u(10)■+■12(7)(10)2
3 Evaluate■and■express■the■answer■in■the■correct■units.
500■=■10u■+■350150■=■10u
u■=■15The■initial■speed■of■the■car■is■15■■m/s.
remember
1.■ A■pronumeral■is■a■letter■or■a■group■of■letters■that■is■used■in■place■of■a■number.2.■ The■coeffi■cient■of■a■term■is■the■number■in■front■of■the■pronumeral(s).3.■ An■expression■is■a■group■of■terms■separated■by■+■or■-■signs.4.■ A■term■that■does■not■contain■a■pronumeral■part■is■called■a■constant.■That■is,■the■term■is■
independent■of■the■variable(s).5.■ When■writing■expressions,■think■about■which■operations■are■being■used,■and■the■order■
in■which■they■occur.6.■ If■pronumerals■are■not■given■in■a■question,■choose■an■appropriate■letter■to■use.7.■ To■evaluate■(fi■nd■the■value■of)■an■algebraic■expression,■substitute■the■pronumerals■with■
their■known■values.8.■ Rather■than■showing■the■multiplication■signs,■it■is■common■in■mathematics■to■write■the■
substituted■values■in■brackets.9.■ An■equation■is■a■mathematical■sentence■that■puts■two■expressions■equal■to■each■other.
using pronumeralsfluenCY
1 Find■the■coeffi■cient■of■each■of■the■following■terms.a 3x b 7a c -2m d -8q e w
f -n gx3
hy2
i - at4
j - r2
9
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exerCise
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3 7 -2 -8 1
-113
1
2- 1
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2 In■each■of■the■following■expressions■state■the■coeffi■cient■of■the■x■term.a 6x■-■3y b 5■+■7x c 5x2■+■3x■-■2 d -7x2■-■2x■+■4e 3x■-■2x2 f -9x2■-■2x g 5x2■+■3■-■7x h -11x■+■5■-■2x2
i 22
2axx
x− + j 1■-23
2xbx− k x x2
657 4
+ + l x3■+■x■+■4
m -3x■-■4bx■+■6 n 4cx2■-■2x■+■4ax o 2x2■-■5
3 We 1 ■Answer■the■following■for■each■expression■below. i State■the■number■of■terms. ii State■the■coeffi■cient■of■the■fi■rst■term. iii State■the■constant■term■(if■there■is■one). iv State■the■term■with■the■smallest■coeffi■cient.
a 5x2■+■7x■+■8 b -9m2■+■8m■-■6■c 5x2y■-■7x2■+■8xy■+■5 d 9ab2■-■8a■-■9b2■+■4e 11p2q2■-■4■+■5p■-■7q■-■p2 f -9p■+■5■-■7q2■+■5p2q■+■qg 4a■-■2■+■9a2b2■-■3ac h 5s■+■s2t■+■9■+■12t■-■3ui -m■+■8■+■5n2m■+■m2■+■2n j 7c2d■+■5d2■+■14■-■3cd2■-■2e
4 We2 ■Write■algebraic■expressions■for■each■of■the■following:a a■number■2■more■than■p b a■number■7■less■than■qc 2■is■added■to■3■times■p d 7■is■subtracted■from■9■times■qe 4■times■p■is■subtracted■from■10 f 5■is■subtracted■from■2■times■pg the■sum■of■p■and q h the■difference■between■p■and■qi the■product■of3■and■p■is■added■to■q j the■product■of■2■and■q■is■subtracted■from■pk the■product■of■p■and■q l 4■times■the■product■of■p■and■qm the■sum■of■2■times■p■and■3■times■q n 3■times■p■is■subtracted■from■2■times■qo p■is■divided■by■two■times■q p 3q■is■divided■by■p.
5a■ ■ mC ■There■are■27■students■in■the■classroom■and■x■students■are■called■out■to■see■the■principal.■The■number■remaining■in■the■room■is:
A 27x B 27■-■x C x■-■27 D 27■+■x E27x
b If■y■people■enter■a■shop■where■there■are■11■customers■and■2■sales■assistants,■the■number■of■people■in■the■shop■is:
A y■+■11 B y■-■13 C 13y D 13■+■y E13y
c If■a■packet■of■Smarties■contains■p■Smarties,■and■they■are■to■be■divided■up■among■4■people,■the■number■of■Smarties■each■person■receives■is:
Ap
4B 4p C 4■+■p D p■-■4 E 4■-■p
d If■a■T-shirt■costs■n■dollars,■ten■T-shirts■would■cost:
A n■+■10 B 10n Cn
10D 10n■+■10 E 10■-■n
6 We3 ■Find■the■value■of■the■following■expressions■if■x■=■2,■y■=■-1■and■z■=■3.a 2x b 3xy
c 2y2z d 14x
e 6(2x■+■3y■–■z) f x2■–■y2■+■xyz
7 If■x■=■4■and■y■=■-3,■evaluate■the■following■expressions.a 4x■+■3yb 3xy■-■2x■+■4c x2■-■y2
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Activity 3-A-1Pronumeral memory
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Activity 3-A-2Language of algebra
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Activity 3-A-3Reviewing algebra
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6 7 3 -23 -2 -7 -11
−12
−23
14
1
-3 -2 0
p■+■2 q■-■7
3p■+■2 9q■-■7
10■-■4p 2p■-■5
p■+■q p■-■q
3p■+■q p■-■2q
pq 4pq
2p■+■3q 2q■-■3p
✔
✔
✔
✔
4 -6
6 7
-12 -3
7
-40
7
3 a i■ 3 ii 5 iii 8 iv 5x2
b i■ 3 ii -9 iii -6 iv -9m2
c i■ 4 ii 5 iii 5 iv -7x2
d i■ 4 ii 9 iii 4 iv -9b2
e i■ 5 ii 11 iii -4 iv -7q f i■ 5 ii -9 iii 5 iv -9p
g i■ 4 ii 4 iii -2 iv -3ac h i■ 5 ii 5 iii 9 iv -3u i i■ 5 ii -1 iii 8 iv -m j i 5 ii 7 iii 14 iv -3cd2
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58 maths Quest 9 for the Australian Curriculum
unDerstAnDing
8 We4 and 5 ■The■formula■for■the■voltage■in■an■electrical■circuit■can■be■found■using■the■formula■known■as■Ohm’s■Law:■V■=■IR where■I■=■current■in■amperes,■R■=■resistance■in■ohms■and■V■=■voltage■in■volts.■a Calculate■V■when:
i I■=■4■amperes,■R■=■8■ohms■ ii■ I■=■25■amperes,■R■=■10■ohms.b Calculate■R■when:
i V =■100■volts,■I■=■25■amperes■ ii■ V =■90■volts,■I■=■30■amperes. 9 Evaluate■each■of■the■following■by■substituting■the■given■values■into■each■formula.
a If■A■=■bh,■fi■nd■A■when■b■=■5■and■h■=■3.
b If■d■=mv
,■fi■nd■d■when■m■=■30■and■v■=■3.
c If■A■=■12 ■xy,■fi■nd■A■when■x■=■18■and■y■=■2.
d If■A■=■12■(a■+■b)h,■fi■nd■A■when■h■=■10,■a■=■7■and■b■=■2.
e If■V■= AH3
,■fi■nd■V■when■A■=■9■and■H■=■10.
f If■v■=■u■+■at,■fi■nd■v■when■u■=■4,■a■=■3.2■and■t■=■2.1.g If■t■=■a■+■(n■-■1)d,■fi■nd■t■when■a■=■3,■n■=■10■and■d■=■2.
h If■A■=■12■(x■+■y)h,■fi■nd■A■when■x■=■5,■y■=■9■and■h■=■3.2.
i If■A■=■2b2,■fi■nd■A■when■b■=■5.j If■y■=■5x2■-■9,■fi■nd■y■when■x■=■6.k If■y■=■x2■-■2x■+■4,■fi■nd■y■when■x■=■2.l If■a■=■-3b2■+■5b■-■2,■fi■nd■a■when■b■=■4.m If■s■=■ut■+■1
2■at2,■fi■nd■s■when■u■=■0.8,■t■=■5■and■a■=■2.3.
n If■F■=mp
r2,■fi■nd■F■correct■to■2■decimal■places,■when■m■=■6.9,■p■=■8■and■r■=■1.2.
o If■C■=■p■d,■fi■nd■C■correct■to■2■decimal■places■if■d■=■11.
reAsoning
10 a■ ■The■area■of■a■triangle■is■given■by■the■formula■A■=■1
2■bh,■where■b■is■the■length■of■
the■base■and■h is■the■perpendicular■height■of■the■triangle.■
i Show■that■the■area■is■12■cm2■when■b■=■6■■cm■and■h■=■4■■cm.
ii What■is■h■if■A■=■24■■cm2■and■b■=■4■■cm?
b The■formula■to■convert■degrees■Fahrenheit■(F )■to■degrees■Celsius■(C )■is■C■=■5
9(F■-■32).
i Find■C■when■F■=■59.■ ■ii Show■that■when■Celsius■(C )■is■15,■
Fahrenheit■(F )■is■59.c The■length■of■the■hypotenuse■of■a■right-
angled■triangle■(c)■can■be■found■using■
the■formula,■c a b= +2 2 ,■where■a■and■b■are■the■lengths■of■the■other■two■sides.■
i Find■c■when■a■=■3■and■b■=■4. ii Find■b■if■c■=■13■and■a■=■5.
32 250
4 3
15
10
18
45
30
10.7221
22.4
50171
4-30
32.75
38.33
34.56
12■■cm2
12■■cm
15
Answers■will■vary.
512
number AnD AlgebrA • pAtterns AnD AlgebrA
59Chapter 3 Algebra
d If■the■volume■of■a■prism■(V)■is■given■by■the■formula■V■=■AH,■where■A■is■the■area■of■the■cross-section■and■H■is■the■height■of■the■prism,■determine:
i V■when■A■=■7■■cm2■and■H■=■9■■cm■ ii H■when■V■=■120■■cm3■and■
A■=■30■■cm2.e Using■E■=■F■+■V■-■2■where■F■is■the■
number■of■faces■on■a■prism,■E■is■the■number■of■edges■and■V■is■the■number■of■vertices,■calculate:
i E■if■F■=■5■and■V■=■7 ii F■if■E■=■10■and■V■=■2.f The■kinetic■energy■(E)■of■an■object■is■
found■by■using■the■formula■E■=■12■mv2■
where■m■is■the■mass■and■v■is■the■velocity■of■the■object.
i Determine■E■when■m■=■3■and■v■=■3.6.
ii Determine■m■whenE■=■25■and■v■=■5.
g The■volume■of■a■cylinder■(v)■is■given■by■v■=■p r 2h,■where■r■is■the■radius■in■centimetres■and■h■is■the■height■of■the■cylinder■in■centimetres.■
i Determine■v■correct■to■2■decimal■places■if■r■=■7■and■h■=■3. ii Determine■h■correct■to■2■decimal■places■if■v■=■120■and■r■=■2.h The■surface■area■of■a■cylinder■(S)■is■given■by■S■=■2pr(r■+■h)■where■r■is■the■radius■of■the■
circular■end■and■h■is■the■height■of■the■cylinder.■ i Calculate■S■(to■2■decimal■places)■
if■r■=■14■and■h■=■5. ii Show■that■for■a■cylinder■of■surface■area■240■and■
radius■5■units,■the■height■is■2.64,■correct■to2■decimal■places.
Algebra in worded problems■■ An■important■skill■in■algebra■is■to■be■able■to■convert■worded■questions,■or■sentences,■into■algebraic■expressions.
■■ The■fi■rst■step■in■converting■a■worded■question■into■an■algebraic■expression■is■to■identify■the■unknown■quantities.
■■ Identify■the■coeffi■cients,■the■constants■and■the■arithmetic■operations■that■connect■them■to■form■an■algebraic■expression.
■■ Assign■a■pronumeral(s)■to■the■unknown■quantity■(or■quantities).■■ Defi■ne■the■pronumeral(s)■in■terms■of■the■quantity■it■represents.
Convert the following sentences into algebraic expressions.a If it takes 8 minutes to iron a single shirt, how long would it take to iron all of Alan’s shirts?b Brenda has $5 more than Camillo. How much money does Brenda have?c In a game of Aussie rules, David kicked 3 more goals than he kicked behinds. How many points
did David score? (1 goal scores 6 points; 1 behind scores 1 point.)
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refleCtion
Which letters (pronumerals) should you avoid using when writing algebraic expressions?
3b
WorkeD exAmple 6
63■cm3
4■cm
1010
19.44
2
461.81
9.55
1671.33
2.64
number AnD AlgebrA • pAtterns AnD AlgebrA
60 maths Quest 9 for the Australian Curriculum
think Write
a 1 Read■the■question■carefully■and■identify■any■unknown■quantities.■The■number■of■Alan’s■shirts■is■unknown.
a
2 Use■a■pronumeral■for■the■unknown■quantity. Let■n■=■the■number■of■Alan’s■shirts.
3 The■total■time■taken■is■the■time■taken■to■iron■1■shirt■multiplied■by■the■number■of■shirts.
The■total■time■is■8■ì■n =■8n.
b 1 Read■the■question■carefully■and■identify■any■unknown■quantities.■The■amount■of■money■that■Camillo■and■Brenda■each■has■is■unknown.
b
2 Use■pronumerals■for■the■unknown■quantities.
Let■b■=■the■amount■of■money■(in■$)■Brenda■has.Let■c■=■the■amount■of■money■(in■$)■Camillo■has.
3 To■find■the■amount■of■money■Brenda■has■we■must■add■$5■to■the■amount■that■Camillo■has.
b■=■c■+■5Brenda■has■$■(c+5).
c 1 Read■the■question■carefully■and■identify■any■unknown■quantities.■The■number■of■goals■and■behinds■kicked■by■David■is■unknown.
c
2 Use■pronumerals■for■the■unknown■quantities.■We■need■only■1■pronumeral■because■there■were■3■fewer■behinds■kicked■than■goals.
Let■g■=■the■number■of■goals■that■David■kicked.The■number■of■behinds■kicked■was■g■-■3.
3 One■goal■is■worth■6■points,■so■multiply■the■number■of■goals■by■6.■One■behind■is■worth■1■point.
Number■of■points■from■goals■=■6■ì■g=■6g■
Number■of■points■from■behinds■=■g■-■3
4 Add■the■points■from■goals■and■behinds■to■find■the■total■points■scored.
The■number■of■points■scored■=■6g■+■g■-■3=■7g■-■3.
■■ To■check■the■reasonableness■of■the■answer■obtained,■substitute■the■values■into■the■original■expression■or■equation.
If■n■=■2,■8n■=■8■ì■2 =■16.
remember
1.■ If■pronumerals■or■variables■are■not■given■in■a■question,■choose■an■appropriate■letter■to■use.2.■ The■first■step■in■converting■a■worded■question■into■an■algebraic■expression■is■to■identify■
any■unknowns■and■assign■a■pronumeral■to■each.3.■ Worded■questions■need■to■be■read■carefully■so■that■you■can■decide■where■to■place■the■
pronumerals,■coefficients■and■constants■in■an■expression.4.■ Check■to■see■if■an■algebraic■expression■is■reasonable■by■substituting■values■for■the■
pronumerals.
number AnD AlgebrA • pAtterns AnD AlgebrA
61Chapter 3 Algebra
Algebra in worded problemsfluenCY
1 Jacqueline■studies■5■more■subjects■than■Helena.■How■many■subjects■does■Jacqueline■study■if:a Helena■studies■6■subjects?b Helena■studies■x■subjects?c Helena■studies■y■subjects?
2 Dianne■and■Angela■walk■home■from■school■together.■Dianne’s■home■is■2■km■further■from■school■than■Angela’s■home.■How■far■does■Dianne■walk■if■Angela’s■home■is:■a 1.5■km■from■school?b x■km■from■school?
3 Lisa■watched■television■for■2.5■hours■today.■How■many■hours■will■she■watch■tomorrow■if■she■watches:a 1.5■hours■more■than■she■watched■today?b t■hours■more■than■she■watched■today?c y■hours■fewer■than■she■watched■today?
unDerstAnDing
4 We6 ■Convert■the■following■sentences■into■algebraic■expressions.a If■it■takes■10■minutes■to■iron■a■single■shirt,■how■long■would■it■take■to■iron■all■of■Anthony’s■
shirts■if■Anthony■has■n■shirts?b Ross■has■30■dollars■more■than■Nick.■If■Nick■has■N■dollars,■how■much■money■does■
Ross■have?c In■a■game■of■Aussie■rules,■Luciano■kicked■4■more■goals■than■he■kicked■behinds.■How■
many■points■did■Luciano■score■if■g■is■the■number■of■goals■kicked?■(Remember:■1■goal■scores■6■points,■1■behind■scores■1■point.)
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3b
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Activity 3-B-1Algebra in words
doc-3981
Activity 3-B-2Using algebra in
worded problemsdoc-3982
Activity 3-B-3Applying algebra in
worded problemsdoc-3983
inDiViDuAl pAthWAYs
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11
x■+■5y■+■5
3.5■km(x■+■2)■km
42.5■+■t
2.5■- y
10n,■where■n■=■number■of■shirts
N■+■30,■where■N■=■the■number■of■Nick’s■dollars
7g■-■4,■where■g■=■goals■scored
number AnD AlgebrA • pAtterns AnD AlgebrA
62 maths Quest 9 for the Australian Curriculum
5 Jeff■and■Chris■play■Aussie■Rules■football■for■opposing■teams,■and■Jeff’s■team■won■when■the■two■teams■played■one■another.■a How■many■points■did■Jeff’s■team■score■if■
they■kicked: i 14■goals■and■10■behinds?ii x■goals■and■y■behinds?
b How■many■points■did■Chris’s■team■score■if■his■team■kicked:
i 10■goals■and■6■behinds? ii p■goals■and■q■behinds?c How■many■points■did■Jeff’s■team■win■by■if: i Chris’s■team■scored■10■goals■and■
6■behinds,■and■Jeff’s■team■scored■14■goals■and■10■behinds?
ii Chris’s■team■scored■p■goals■and■q■behinds,■and■Jeff’s■team■scored■x■goals■and■y■behinds?
6 Yvonne’s■mother■gives■her■x■dollars■for■each■school■subject■she■passes.■If■she■passes■y■subjects,■how■much■money■does■she■receive?
7 Brian■buys■a■bag■containing■x Smarties.a If■he■divides■them■equally■among■n■people,■how■
many■does■each■person■receive?b If■he■keeps■half■the■Smarties■for■himself■and■
divides■the■remaining■Smarties■equally■among■n■people,■how■many■does■each■person■receive?
8 A■piece■of■licorice■is■30■■cm■long.a If■David■cuts■d■■cm■off,■how■much■licorice■
remains?b If■David■cuts■off■1
4■of■the■remaining■licorice,■how■much■licorice■has■been■cut■off?
c How■much■licorice■remains■now?
reAsoning
9 One-quarter■of■a■class■of■x■students■plays■tennis■on■the■weekend.■One-sixth■of■the■class■plays■tennis■and■swims■on■the■weekend.a Write■an■expression■to■represent■the■number■of■
students■playing■tennis■on■the■weekend.b Write■an■expression■to■represent■the■number■of■
students■playing■tennis■and■swimming■on■the■weekend.
c Show■that■the■number■of■students■playing■only■tennis■on■the■weekend■is■
x12
.
10 During■a■24-hour■period,■Vanessa■uses■her■computer■for■c■hours.■Her■brother■Darren■uses■it■for■1
7■of■the■
remaining■time.■
a For■how■long■does■Darren■use■the■computer?b Show■that■the■total■number■of■hours■that■Vanessa■and■Darren■use■the■computer■during■a■
24-hour■period■can■be■expressed■as■6 247
x + .
946x■+■y
666p■+■q
28
6x■+■y■-■6p■-■q
xy■dollars
(30■-■d )■cm30
4− d cm
x x x4 6 12
− =
x4x6
(c■+■24
7− c
)■hours
247− c
■hours
3 34
( )3 3( )3 30( )0( )−( )d( )d( ) cm
xn2
xn
number AnD AlgebrA • pAtterns AnD AlgebrA
63Chapter 3 Algebra
11 Marty■had■a■birthday■party■last■weekend■and■invited■n■friends■where■n■≥■24.■The■table■below■indicates■the■number■of■friends■at■Marty’s■party■at■the■specified■times■during■the■evening.■They■all■left■the■party■by■11■pm.a How■many■people■arrived■between■7.00■pm■and■7.30■pm?b Between■which■times■were■the■most■friends■present■at■the■party?c How■many■friends■were■invited■but■did■not■arrive?d How■many■friends■were■invited■in■total?e Between■which■times■did■the■most■friends■arrive?f What■assumptions■have■been■made■in■the■previous■answers?
Time Number of friends
■ 7.00■pm n■–■24
■ 7.30■pm n■–■23
■ 8.00■pm n■–■8
■ 8.30■pm n■–■5
■ 9.00■pm n■–■5
■ 9.30■pm n■–■7
10.00■pm n■–■12
10.30■pm n■–■18
11.00■pm n■–■24
simplifying algebraic expressions■■ In■this■section■the■methods■of■simplifying■algebraic■expressions■will■be■reviewed.
Addition and subtraction of like terms■■ Like terms■contain■the■same■pronumeral■parts.
For each of the following terms, select those terms listed in brackets that are like terms.a 4y (y, -y, 4x, 4xy, -4y)b 5xy (-5xy, 5x, 5yx, 5xz, -xy, -x2y, 5(xy)2)c -6abc (-6bca, -6abd, -6a2bc, -2acb, -2ac2b)d -7q2b3e4 (-7q2b2e2, -6b3e4q2, 6q2e4b3, 7q4b3e4, -7q2b2e2)
think Write
a The■pronumeral■part■of■4y■is■y.■Check■the■list■for■terms■with■the■same■pronumeral■part.
a Like■terms:■y,■-y,■-4y
b The■pronumeral■part■of■5xy■is■xy.■Check■the■list■for■terms■with■the■same■pronumeral■part.
b Like■terms:■-5xy,■5yx,■-xy
c The■pronumeral■part■of■-6abc■is■abc.■Check■the■list■for■terms■with■the■same■pronumeral■part.
c Like■terms:■-6bca,■-2acb
d The■pronumeral■part■of■-7q2b3e4■is■q2b3e4.■Check■the■list■for■terms■with■the■same■pronumeral■part.
d Like■terms:■-6b3e4q2,■6q2e4b3
■■ When■like■terms■appear■in■an■expression,■they■can■be■collected■(added■or■subtracted).■■■ Always■take■note■of■the■sign■in■front■of■the■term.
refleCtion
Why is it important to define pronumerals or variables in terms of what they represent?
3C
WorkeD exAmple 7
18.30■pm■and■9.30■pm
524
None■of■Marty’s■friends■left■and■then■returned;■also,■nobody■arrived■who■hadn’t■been■invited.
7.30■pm■and■8.00■pm
number AnD AlgebrA • pAtterns AnD AlgebrA
64 maths Quest 9 for the Australian Curriculum
Simplify the following expressions.a 6x + 5y - 4x + 2y b 9a2b - 3ab2 + 2ab c 6a2 + 9b + 7b2 - 5b d 12 - 4a2b + 2 - 2ba2 e -12 - 4c2 + 10 + 2c2
think Write
a 1 Write■the■expression. a 6x■+■5y■-■4x■+■2y
2 Identify■and■collect■the■like■terms. =■6x■-■4x■+■5y■+■2y
3 Simplify■by■adding■or■subtracting■like■terms. =■2x■+■7y
b 1 Write■the■expression. b 9a2b■-■3ab2■+■2ab
2 Identify■the■like■terms.■There■are■none! Cannot■be■simplifi■ed.
c 1 Write■the■expression. c 6a2■+■9b■+■7b2■-■5b
2 Identify■and■collect■the■like■terms. =■6a2■+■9b■-■5b■+■7b2
3 Simplify. =■6a2■+■4b■+■7b2
d 1 Write■the■expression. d 12■-■4a2b■+■2■-■2ba2
2 Identify■and■collect■the■like■terms. =■12■+■2■-■4a2b■-■2ba2
3 Simplify. =■14■-■6a2b
e 1 Write■the■expression. e -12■-■4c2■+■10■+■2c2
2 Identify■and■collect■the■like■terms. =-12■+■10■-■4c2■+■2c2
3 Simplify. =-2■-■2c2
multiplication and division■■ When■multiplying■and■dividing■algebraic■terms,■it■is■not■necessary■to■have■like■terms.■■■ To■multiply■or■divide■algebraic■terms,■fi■nd■the■product■or■quotient■of■the■coeffi■cients■separately■to■the■pronumerals.
Simplify the following.a 4aì2bìa b 7ax ì -6bx ì -2abx c
410
xyyz
d 8ab ó 16a2b
think Write
a 1 Write■the■algebraic■expression. a 4a■ì■2b■ì■a
2 Rearrange,■writing■the■coeffi■cients■fi■rst. =■4■ì■2■ì■a■ì■a■ì■b
3 Multiply■the■coeffi■cients■and■pronumerals■separately.
=■8■ì■a2■ì■b=■8a2b
b 1 Write■the■algebraic■expression. b 7ax■ì■-6bx■ì■-2abx
2 Rearrange,■writing■the■coeffi■cients■fi■rst. =■7■ì■-6■ì■-2■ì■a■ì■a■ì■x■ì■x■ì■x■ì■b■ì■b
3 Multiply■the■coeffi■cients■and■pronumerals■separately.■The■simplifi■ed■term■is■often■written■with■the■pronumerals■in■alphabetical■order.
=■84■ì■a2■ì■x3■ì■b2
=■84a2b2x3
WorkeD exAmple 8
WorkeD exAmple 9
number AnD AlgebrA • pAtterns AnD AlgebrA
65Chapter 3 Algebra
c 1 Write■the■term. c4
10xyyz
2 Cancel■4■and■10■(common■factor■2).■Cancel■y■from■the■numerator■and■the■denominator.
= 2
5
4
10
xy
yz
=■25
xz
d 1 Write■the■algebraic■expression■and■express■as■a■fraction.■The■term■a2■means■aa.
d 8ab■ó■16a2b
=■8
16 2
ab
a b
=■8
16abaab
2 Cancel■8■and■16■(common■factor■8).■Cancel■a■and■b■from■the■numerator■and■the■denominator.
=■1
2
8
16
a b
a a b
=■1
2a
remember
1.■ Like■terms■contain■the■same■pronumeral■parts.2.■ When■like■terms■appear■in■an■expression,■they■can■be■added■or■subtracted.3.■ When■multiplying■and■dividing■algebraic■terms,■it■is■not■necessary■to■have■like■terms.4.■ For■multiplication,■we■can■multiply■the■coeffi■cients■(number■parts)■and■the■pronumeral■
parts■separately.5.■ A■division■problem■should■be■expressed■as■a■fraction.6.■ For■division,■always■check■to■see■if■the■fraction■can■be■simplifi■ed■by■cancelling■the■
numerator■and■denominator■by■any■common■factors.
simplifying algebraic expressionsfluenCY
1 We7 ■For■each■of■the■following■terms,■select■those■terms■listed■in■brackets■that■are■like■terms.a 6ab■ (7a,■8b,■9ab,■-ab,■4a2b2)b -x■ (3xy,■-xy,■4x,■4y,■-yx)c 3az■ (3ay,■-3za,■-az,■3z2a,■3a2z)d x2■ (2x,■2x2,■2x3,■-2x,■-x2)e -2x2y■ (xy,■-2xy,■-2xy2,■-2x2y,■-2x2y2)f 3x2y5■ (3xy,■3x5y2,■3x4y3,■-x2y5,■-3x2y5)g 5x2p3w5■ (-5x3w5p3,■p3x2w5,■5xp3w5,■-5x2p3w5,■w5p2x3)h -x2y5z4■ (-xy5,■-y2z5x4,■-x■+■y■+■z,■4y5z4x2,■-2x2z4y5)
2 We8 Simplify■the■following■expressions.a 5x■+■2x b 3y■+■8y c 7m■+■12md 13q■-■2q e 17r■-■9r f -x■+■4xg 5a■+■2a■+■a h 9y■+■2y■-■3y i 7x■-■2x■+■8x
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Activity 3-C-1Reviewing algebraic
operationsdoc-3984
Activity 3-C-2Simplifying algebraic
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Activity 3-C-3Applying algebraic
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inDiViDuAl pAthWAYs
9ab,■-ab
4x
-3za,■-az
2x2,■-x2
-2x2y-x2y5,■-3x2y5
p3x2w5,■-5x2p3w5
4y5z4x2,■-2x2z4y5
7x 11y 19m11q 8r 3x
8a 8y 13x
number AnD AlgebrA • pAtterns AnD AlgebrA
66 maths Quest 9 for the Australian Curriculum
j 14p■-■3p■+■5p k 2q2■+■7q2 l 5x2■-■2x2
m 6x2■+■2x2■-■3y n 3m2■+■2n■-■m2 o 9x2■+■x■-■2x2
p 9h2■-■2h■+■3h■+■9 q -2g2■-■4g■+■5g■-■12 r -5m2■+■5m■-■4m■+■15s 12a2■+■3b■+■4b2■-■2b t 6m■+■2n2■-■3m■+■5n2 u 3xy■+■2y2■+■9yxv 3ab■+■3a2b■+■2a2b■-■ab w 9x2y■-■3xy■+■7yx2 x 4m2n■+■3n■-■3m2n■+■8ny -3x2■-■4yx2■-■4x2■+■6x2y z 4■- 2a2b■-■ba2■+■5b - 9a2
3 mC Simplify■the■following■expressions.a 18p■-■19p
A p B -p C p2
D -1 E 1b 5x2■-■8x■+■6x■-■9
A 3x■-■9 B 3x2■-■9 C 5x2■+■2x■-■9D 5x2■-■2x■-■9 E -3x■-■9
c 12a■-■a■+■15b■-■14bA 11a■+■b B 12 C 11a■-■bD 13a■+■b E 12a■+■b
d -7m2n■+■5m2■+■3■-■m2■+■2m2nA -9m2n■+■4m2■+■3D -5m2n■+■4m2■+■3
B -9m2n■+■8E -5m2n■+■3
C -5m2n■-■4m2■+■3
4 We9a, b Simplify■the■following.a 3m■ì■2n b 4x■ì■5y c 2p■ì■4qd 5x■ì■-2y e 3y■ì■-4x f -3m■ì■-5ng 5a■ì■2a h 4y■ì■5y i 5p■ì■pj m■ì■7m k 3mn■ì■2p l -6ab■ì■bm -5m■ì■-2mn n -6a■ì■3ab o -3xy■ì■-5xy■ì■2xp 4pq■ì■-p■ì■3q2 q 4c■ì■-7cd■ì■2c r -3a2■ì■-5ab3■ì■2ab4
5 We9c, d ■Simplify■the■following.
a62x
b93m
c12
6y
d82m
e 12m■ó■3 f 14x■ó■7
g -21x■ó■3 h -32m■ó■8 i48m
j618
xk
818mn
nl
1612
xyy
m2
10mn
6
12 2
ab
a bo
2814
xyzx
p 704
2abb
q 28
2x yzxz
r -7xy2z2■ó■11xyz
6 Simplify■the■following.a 5x■ì■4y■ì■2xy b 7xy■ì■4ax■ì■2y c x■ì■4xy■ì■3yx
d6
12
2
2
x y
ye
−15
12
2
2 2
x ab
b xf
2 3 2
3 2
p q
p qg -4a■ì■-5ab2■ì■2a h -a■ì■4ab■ì■2ba■ì■b i 2a■ì■2a■ì■2a■ì■2a
unDerstAnDing
7 Jim■buys■m■pens■at■p■cents■each■and■n■books■at■q■dollars■each.a How■much■does■he■spend■in: i dollars?■ ii■ cents?b What■is■his■change■from■$20?
eBookpluseBookplus
Digital docSkillSHEET 3.5
doc-6126
eBookpluseBookplus
Digital docSkillSHEET 3.4
doc-6125
eBookpluseBookplus
Digital docSkillSHEET 3.18
doc-6140
eBookpluseBookplus
Digital docSkillSHEET 3.19
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SkillSHEET 3.6doc-6127
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Digital docSkillSHEET 3.7
doc-6128
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Digital docWorkSHEET 3.2
doc-6142
16p 9q2 3x2
8x2■-■3y 2m2■+■2n 7x2■+■x
2q■ -2g2■+■g■-12
r -5m2■+■m■+■15 s 12a2■+■b■+■4b2
t 3m■+■7n2
u 12xy■+■2y2
v 2ab■+■5a2b w 16x2y■-■3xy x m2n■+■11n y -7x2■+■2x2y
✔
✔
✔
✔
6mn 20xy 8pq
-10xy -12xy 15mn10a2 20y2 5p2
7m2 6mnp -6ab2
10m2n -18a2b 30x3y2
-12p2q3 -56c3d 30a4b7
3x 3m 2y
4m 4m 2x
-7x -4m m2
x3
49m 4
3x
15m
12a
2yz
352ab xy
4−711
yz
40x2y2■ 56ax2y2■ 12x3y2
xy
2
2
−54
ab
2
40a3b2 -8a3b3 16a4
(0.01mp■+■nq)■dollars (mp■+■100nq)■cents$20■-■(0.01mp■+■nq)
9h2■+■h■+■9
-3a2b■-■9a2■+■5b■+4
number AnD AlgebrA • pAtterns AnD AlgebrA
67Chapter 3 Algebra
8 At■a■local■discount■clothing■store■4■shirts■and■3■pairs■of■trousers■cost■$138.If■a■pair■of■trousers■cost■2.5■times■as■much■as■a■shirt,■determine■the■cost■of■each.
reAsoning
9 Class■9A■were■given■an■algebra■test.■One■of■the■questions■is■shown■below:
Simplify■the■following■expression■ 32
46
7ab ac
bc× ×
■Sean■who■is■a■student■in■class■9A■wrote■his■answer■as■■12
127
aabcb
c× .■Explain■why■Sean’s■answer■is■
incorrect,■and■write■down■the■correct■answer.
expanding bracketsexpanding single brackets
■■ Expansion■means■to■multiply■everything■inside■the■brackets■by■what■is■directly■outside■the■brackets.■
■■ This■involves■applying■the■Distributive Law.■■ Recall■a(b■+■c)■=■a■ì■b■+■a■ì■c =■ab■+■ac
■■ The■Distributive■Law■can■be■illustrated■using■the■area■of■a■rectangle.■■■ If■one■side■of■the■rectangle■has■a■length■of■(b■+■c)■units■and■the■other■side■has■a■length■of■a■units,■then■it■can■be■seen■that■the■total■area■of■the■rectangle■is■a(b■+■c)■=■ab■+■ac.
b + cc
ac
b
aba
■■ This■method■can■be■confirmed■with■numbers.For■example■9(5■+■4)■=■9■ì■5■+■9■ì■4
=■45■+■36 =■81
5
9 ì 5 = 45 9 ì 4 = 36
4
9
Expand the following expressions.a 5(x + 3) b 8(x - y) c -a(x - y)
think Write
a 1 Write■the■expression. a 5(x■+■3)
2 Expand■the■brackets. =■5■ì■x■+■5■ì■3
3 Simplify. =■5x■+■15
refleCtion
Is the expression ab the same as ba ? Explain.
3D
WorkeD exAmple 10
Explanations■will■vary.■Correct■answer■is■7a2c2.
Shirt■=■$12■eachTrousers■=■$30■each
number AnD AlgebrA • pAtterns AnD AlgebrA
68 maths Quest 9 for the Australian Curriculum
b 1 Write■the■expression. b 8(x■-■y)
2 Expand■the■brackets. =■8■ì■x■+■8■ì■-y
3 Simplify. =■8x■-■8y
(Remember■that■a■positive■term■multiplied■by■a■negative■term■gives■a■negative■term.)
c 1 Write■the■expression. c -a(x■-■y)
2 Expand■the■brackets. =■-a■ì■x■-■a■ì■-y
3 Simplify. =■-ax■+■ay
(Remember■that■a■negative■term■multiplied■by■a■negative■term■gives■a■positive■term.)
Note:■It■doesn’t■matter■what■is■immediately■outside■the■brackets.■It■may■be■a■number■or■a■pronumeral■or■both.■The■following■expansions■are■a■little■more■complex,■but■the■Distributive■Law■is■applied■in■the■same■manner.
Expand each of the following.a 5x(6y - 7z) b -4y(2x + 3w) c x(2x + 3y)
think Write
a 1 Write■the■expression. a 5x(6y■-■7z)
2 Expand■the■brackets. =■5x■ì■6y■+■5x■ì■-7z
3 Simplify. =■30xy■-■35xz
(Multiply■number■parts■and■pronumeral■parts■separately■and■write■pronumerals■for■each■term■in■alphabetical■order.)
b 1 Write■the■expression. b -4y(2x■+■3w)
2 Expand■the■brackets. =■-4y■ì■2x■-■4y■ì■3w
3 Simplify. =■-8xy■-■12wy
c 1 Write■the■expression. c x(2x■+■3y)
2 Expand■the■brackets. =■x■ì■2x■+■x■+■3y
3 Simplify. =■2x2■+■3xy
(Remember■that■x■multiplied■by■itself■gives■x2.)
expanding and collecting like terms■■ With■more■complicated■expansions,■like■terms■may■need■to■be■collected■after■the■expansion■of■the■bracketed■part.■
■■ Remember■that■like■terms■contain■the■same■pronumeral■parts.■■■ First■expand■the■brackets,■then■collect■the■like■terms.■■■ This■is■applying■the■BIDMAS■rule■where■one■always■multiplies■(or■divides)■before■adding■or■subtracting.
WorkeD exAmple 11
number AnD AlgebrA • pAtterns AnD AlgebrA
69Chapter 3 Algebra
Expand and simplify by collecting like terms.a 4(x - 4) + 5 b x(y - 2) + 5x c -x(y - z) + 5x d 7x - 6(y - 2x)
think Write
a 1 Write■the■expression. a 4(x■-■4)■+■5
2 Expand■the■brackets. =■4■ì■x■+■4■ì■-4■+■5
3 Simplify. =■4x■-■16■+■5
4 Write■the■answer. =■4x■-■11
b 1 Write■the■expression. b x(y■-■2)■+■5x
2 Expand■the■brackets. =■x■ì■y■+■x■ì■-2■+■5x
3 Simplify. =■xy■-■2x■+■5x
4 Write■the■answer. =■xy■+■3x
c 1 Write■the■expression. c -x(y■-■z)■+■5x
2 Expand■the■brackets. =■-x■ì■y■-■x■ì■-z■+■5x
3 Simplify. =■-xy■+■xz■+■5x
4 There■are■no■like■terms.
d 1 Write■the■expression. d 7x■-■6(y■-■2x)
2 Expand■the■brackets. =■7x■-■6■ì■y■-■6■ì■-2x
3 Simplify. =■7x■-■6y■+■12x
4 Write■the■answer. =■19x■-■6y
expanding two brackets■■ When■expanding■an■expression■that■contains■two■(or■more)■brackets,■the■steps■are■the■same■as■before.Step■1.■ Expand■each■bracket■(working■from■left■to■right).Step■2.■ Collect■any■like■terms■and■simplify.
Expand and simplify the following expressions.a 5(x + 2y) + 6(x - 3y) b -5x( y - 2) + y(x + 3)c 7y(x - 2y) + y2(x + 5) d -5xy(1 + 2y) + 6x( y + 4x)
think Write
a 1 Write■the■expression. a 5(x■+■2y)■+■6(x■-■3y)
2 Expand■each■bracket. =■5■ì■x■+■5■ì■2y■+■6■ì■x■+■6■ì■-3y
3 Simplify. =■5x■+■10y■+■6x■-■18y
4 Write■the■answer. =■11x■-■8y
b 1 Write■the■expression. b -5x(y■-■2)■+■y(x■+■3)
2 Expand■each■bracket. =■-5x■ì■y■-■5x■ì■-2■+■y■ì■x■+■y■ì■3
3 Simplify. =■-5xy■+■10x■+■xy■+■3y
4 Write■the■answer. =■-4xy■+■10x■+■3y
WorkeD exAmple 12
WorkeD exAmple 13
number AnD AlgebrA • pAtterns AnD AlgebrA
70 maths Quest 9 for the Australian Curriculum
c 1 Write■the■expression. c 7y(x■-■2y)■+■y2(x■+■5)
2 Expand■each■bracket. =■7y■ì■x■+■7y■ì■-2y■+■y2■ì■x■+■y2■ì■5
3 Simplify. =■7xy■-14y2■+■xy2■+■5y2
4 Write■the■answer. =■7xy■-■9y2■+■xy2
d 1 Write■the■expression. d -5xy(1■+■2y)■+■6x(y■+■4x)
2 Expand■each■bracket. =■-5xy■ì■1■-■5xy■ì■2y■+■6x■ì■y■+■6x■ì■4x
3 Simplify. =■-5xy■-10xy2■+■6xy■+■24x2
4 Write■the■answer. =■xy■-■10xy2■+■24x2
expanding pairs of brackets■■ In■this■section,■expressions■where■there■are■two■brackets■being■multiplied■together,■such■as■■(x■+■2y)(x■-■3y),■are■explored.■
■■ When■multiplying■expressions■within■brackets,■multiply■each■term■in■the■first■bracket■by■each■term■in■the■second■bracket,■again■applying■the■Distributive■Law.Therefore■ (a■+■b)(c■+d)■=■a(c■+■d)■+■b(c■+■d )
=■ac■+■ad■+■bc■+■bd■■ This■can■be■demonstrated■using■the■area■of■the■rectangle.
c + d
a + b
d
ad
c
aca
bdbcb
■■ This■method■can■be■confirmed■with■numbers.For■example:■(7■+■3)(6■+■2)■=■7(6■+■2)■+■3(6■+■2)
■ =■7■ì■6■+■7■ì■2■+■3■ì■6■+■3■ì■2■ =■42■+■14■+■18■+■6■ =■80
Expand and simplify each of the following expressions.a (x - 5)(x + 3) b (x + 2)(x + 3) c (2x + 2)(2x + 3)
think Write
a 1 Write■the■expression. a (x■-■5)(x■+■3)
2 Expand■by■multiplying■each■of■the■terms■in■the■first■bracket■by■each■of■the■terms■in■the■second■bracket.■Finally■simplify■the■expression■by■collecting■like■terms.
=■x(x■+■3)■-■5(x■+■3)=■x■ì■x■+■x■ì■3■-■5■ì■x■-■5■ì■3=■x2■+■3x■-■5x■-■15■=■x2■–■2x■-■15
7
7 ì 6 = 42
7 ì 2 = 143 ì 2= 6
3 ì 6= 18
3
6
2
WorkeD exAmple 14
number AnD AlgebrA • pAtterns AnD AlgebrA
71Chapter 3 Algebra
b 1 Write■the■expression. b (x■+■2)(x■+■3)
2 Expand■by■multiplying■each■of■the■terms■in■the■fi■rst■bracket■by■each■of■the■terms■in■the■second■bracket.■Finally■simplify■the■expression■by■collecting■like■terms.
=■x(x■+■3)■+■2(x■+■3)=■x■ì■x■+■x■ì■3■+■2■ì■x■+■2■ì■3=■x2■+■3x■+2x■+6■=■x2■+■5x■+■6
c 1 Write■the■expression. c (2x■+2)(2x■+■3)
2 Expand■by■multiplying■each■of■the■terms■in■the■fi■rst■bracket■by■each■of■the■terms■in■the■second■bracket.■Finally■simplify■the■expression■by■collecting■like■terms.
=■2x(2x■+■3)■+2(2x■+■3)=■2x■ì■2x■+■2x■ì■3■+■2■ì■2x■+■2■ì■3=■4x2■+■6x■+■4x■+6■=■4x2■+■10x■+■6
■■ Another■method■that■can■be■used■to■remember■the■expansion■of■two■binomial■factors■is■commonly■known■as■the■‘FOIL’■method.
■■ This■method■is■given■the■name■FOIL■because■the■letters■stand■for:First■ —■ multiply■the■fi■rst■term■in■each■bracket.Outer■ —■ multiply■the■2■outer■terms■of■each■bracket.Inner■ —■ multiply■the■2■inner■terms■of■each■bracket.Last■ —■ multiply■the■last■term■of■each■bracket.
■■ An■interesting■application■of■this■expansion■method■is■its■use■in■simplifying■multiplication.For■example:■98■ì■52■=■(100■-■2)(50■+■2)■ ■ =■100■ì■50■+■100■ì■2■-■2■ì■50■-■2■ì■2■ ■ =■5000■+■200■-■100■-■4■ ■ =■5096(You■can■check■this■answer■by■multiplying■the■two■numbers.)
remember
1.■ Expansion■means■to■multiply■everything■inside■the■brackets■by■what■is■directly■outside■the■brackets.■By■doing■this■we■are■expanding■the■brackets■and■applying■the■Distributive■Law,■which■states■that■a(b■+■c)■=■ab■+■ac.
2.■ After■expanding■brackets,■simplify■by■collecting■any■like■terms.3.■ You■can■use■a■diagrammatic■method■or■FOIL■to■help■you■keep■
track■of■which■terms■are■to■be■multiplied■together.
expanding bracketsfluenCY
1 We 10 Expand■the■following■expressions.a 3(x■+■2) b 4(x■+■3) c 5(m■+■4) d 2(p■+■5)e 4(x■+■1) f 7(x■-■1) g -4(y■+■6) h -5(a■+■1)i -3(p■-■2) j -(x■-■1) k -(x■+■3) l -(x■-■2)m 3(2b■-■4) n 8(3m■-■2) o -6(5m■-■4) p -3(9p■-■5)
2 We 11 Expand■each■of■the■following.a x(x■+■2) b y(y■+■3) c a(a■+■5) d c(c■+■4)e x(4■+■x) f y(5■+■y) g m(7■-■m) h q(8■-■q)i 2x(y■+■2) j 5p(q■+■4) k -3y(x■+■4) l -10p(q■+■9)m -b(3■-■a) n -7m(5■-■n) o -6a(5■-■3a) p -4x(7■-■4x)
(a + b) (c + d)F
O
IL
(a + b) (c + d)F
O
IL
exerCise
3D
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Activity 3-D-1Reviewing expansion
doc-3987
Activity 3-D-2Expanding brackets
doc-3988
Activity 3-D-3Applying bracket
expansiondoc-3989
inDiViDuAl pAthWAYs
3x■+■6 4x■+■12 5m■+■20 2p■+■10
4x■+■4 7x■-■7 -4y■-■24 -5a■-■5
-3p■+■6 -x■+■1 -x■-■3 -x■+■2
6b■-■12 24m■-■16 -27p■+■15
-30m■+■24
x2■+■2x y2■+■3y a2■+■5a c2■+■4c
4x■+■x2 5y■+■y2 7m■-■m2 8q■-■q2
2 i■ 2xy■+■4x j 5pq■+■20p k -3xy■-■12y l -10pq■-■90p m -3b■+■ab n -35m■+■7mn o -30a■+■18a2 p -28x■+■16x2
number AnD AlgebrA • pAtterns AnD AlgebrA
72 maths Quest 9 for the Australian Curriculum
3 We 12 Expand■and■simplify■by■collecting■like■terms.a 2(p■-■3)■+■4 b 5(x■-■5)■+■8c -7(p■+■2)■-■3 d -4(3p■-■1)■-■1e 6x(x■-■3)■-■2x f 2m(m■+■5)■-■3mg 3x(p■+■2)■-■5 h 4y(y■-■1)■+■7i -4p(p■-■2)■+■5p j 5(x■-■2y)■-■3y■-■xk 2m(m■-■5)■+■2m■-■4 l -3p(p■-■2q)■+■4pq■-■1m -7a(5■-■2b)■+■5a■-■4ab n 4c(2d■-■3c)■-■cd■-■5co 6p■+■3■-■4(2p■+■5) p 5■-■9m■+■2(3m■-■1)
4 We 13a Expand■and■simplify■the■following■expressions.a 2(x■+■2y)■+■3(2x■-■y) b 4(2p■+■3q)■+■2(p■-■2q)c 7(2a■+■3b)■+■4(a■+■2b) d 5(3c■+■4d)■+■2(2c■+■d)e -4(m■+■2n)■+■3(2m■-■n) f -3(2x■+■y)■+■4(3x■-■2y)g -2(3x■+■2y)■+■3(5x■+■3y) h -5(4p■+■2q)■+■2(3p■+■q)i 6(a■-■2b)■-■5(2a■-■3b) j 5(2x■-■y)■-2(3x■-■2y)k 4(2p■-■4q)■-■3(p■-■2q) l 2(c■-■3d)■-■5(2c■-■3d)m 7(2x■-■3y)■-■(x■-■2y) n -5(p■-■2q)■-(2p■-■q)o -3(a■-■2b)■-■(2a■+■3b) p 4(3c■+■d)■-■(4c■+■3d)
5 We 13b, c, d Expand■and■simplify■the■following■expressions.a a(b■+■2)■+■b(a■-■3) b x(y■+■4)■+■y(x■-■2)c c(d■-■2)■+■c(d■+■5) d p(q■-■5)■+■p(q■+■3)e 3c(d■-■2)■+■c(2d■-■5) f 7a(b■-■3)■-b(2a■+■3)g 2m(n■+■3)■-■m(2n■+■1) h 4c(d■-■5)■+■2c(d■-■8)i 3m(2m■+■4)■-■2(3m■+■5) j 5c(2d■-■1)■-(3c■+■cd)k -3a(5a■+■b)■+■2b(b■-■3a) l -4c(2c■-■6d)■+■d(3d■-■2c)m 6m(2m■-■3)■-(2m■+■4) n 2p(p■-■4)■+■3(5p■-■2)o 7x(5■-■x)■+■6(x■-■1) p -2y(5y■-■1)■-■4(2y■+■3)
6 mC ■a What■is■the■equivalent■of■3(a■+■2b)■+■2(2a■-■b)?A 5a■+■6b B 7a■+■4b C 5(3a■+■b)D 7a■+■8b E 12a■-■12b
b What■is■the■equivalent■of■-3(x■-■2y)■-(x■-■5y)?A -4x■+■11y B -4x■-■11y C 4x■+■11yD 4x■+■7y E 3x■+■30y
c What■is■the■equivalent■of■2m(n■+■4)■+■m(3n■-■2)?A 3m■+■4n■-■8 B 5mn■+■4m C 5mn■+■10mD 5mn■+■6m E 6mn■-■16m
7 We 14 Expand■and■simplify■each■of■the■following■expressions.a (a■+■2)(a■+■3) b (x■+■4)(x■+■3) c (y■+■3)(y■+■2)d (m +■4)(m■+■5) e (b■+■2)(b■+■1) f (p■+■1)(p■+■4)g (a■-■2)(a■+■3) h (x■-■4)(x■+■5) i (m■+■3)(m■-■4)j (y■+■5)(y■-■3) k (y■-■6)(y■+■2) l (x■-■3)(x■+■1)m (x■-■3)(x■-■4) n (p■-■2)(p■-■3) o (x■-■3)(x■-■1)
8 Use■the■FOIL■technique■to■expand■the■following.a (2a■+■3)(a■+■2) b (3m■+■1)(m■+■2) c (6x■+■4)(x■+■1)d (c■-■6)(4c■-■7) e (7■-■2t)(5■-■t) f (1■-■x)(9■-■2x)g (2■+■3t)(5■-■2t) h (7■-■5x)(2■-■3x) i (5x■-■2)(5x■-■2)
9 Expand■and■simplify■each■of■the■following.a (x■+■y)(z■+■1) b (p■+■q)(r■+■3) c (2x■+■y)(z■+■4)d (3p■+■q)(r■+■1) e (a■+■2b)(a■+■b) f (2c■+■d )(c■-■3d)g (x■+■y)(2x■-■3y) h (4p■-■3q)(p■+■q) i (3y■+■z)(x■+■z)j (a■+■2b)(b■+■c) k (3p■-■2q)(1■-■3r) l (7c■-■2d )(d■-■5)m (4x■-■y)(3x■-■y) n (p■-■q)(2p■-■r) o (5■-■2j)(3k■+■1)
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SkillSHEET 3.5doc-6126
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SkillSHEET 3.8doc-6129
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Digital docSkillSHEET 3.20
doc-6143
2p■-■2 5x■-■17-7p■-■17 -12p■+■3
6x2■-■20x 2m2■+■7m3px■+■6x■-■5 4y2■-■4y■+■7
-4p2■+■13p 4x■-■13y2m2■-■8m■-■4 -3p2■+■10pq■-■1
-30a■+■10ab 7cd■-■12c2■-■5c-2p■-■17 3■-■3m
8x■+■y 10p■+■8q
18a■+■29b 19c■+■22d2m■-■11n 6x■-■11y9x■+■5y -14p■-■8q
-4a■+■3b 4x■-■y5p■-■10q -8c■+■9d
13x■-■19y -7p■+■11q-5a■+■3b 8c■+■d
2ab■+■2a■-■3b 2xy■+■4x■-■2y2cd■+■3c 2pq■-■2p
5cd■-■11c 5ab■-■21a■-■3b5m 6cd■-■36c
6m2■+■6m■-■10 9cd■-■8c-15a2■+■2b2■-■9ab -8c2■+■3d 2■+■22cd
12m2■-■20m■-■4 2p2■+■7p■-■6-7x2■+■41x■-■6 -10y2■-■6y■-■12
✔
✔
✔
a2■+■5a■+■6 x2■+■7x■+■12 y2■+■5y■+■6m2■+■9m■+■20 b2■+■3b■+■2 p2■+■5p■+■4a2■+■a■-■6 x2■+■x■-■20 m2■-■m■-■12
y2■+■2y■-■15 y2■-■4y■-■12 x2■-■2x■-■3x2■-■7x■+■12 p2■-■5p■+■6 x2■-■4x■+■3
8 a■ 2a2■+■7a■+■6 b 3m2■+■7m■+■2
c 6x2■+■10x■+■4 d 4c2■-■31c■+■42
e 35■-■17t■+■2t2
f 9■-■11x■+■2x2
g 10■+■11t■-■6t2
h 14■-■31x■+■15x2
i 25x2■-■20x■+■4
i3y
x■+■
3yz■
+■zx
■+■z
2
jab
■+■a
c■+■
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+■2b
c
k
3p■-
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q■+■
6qr
l7c
d■-■
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7xy■
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2 ■-■
rp■-
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k■-■
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■+■x
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z■+■
y
b
pr■+
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+■qr
■+■3
q
c
2xz■
+■8x
■+■y
z■+■
4y
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r■+■
3p■+
■qr■
+■q
ea2 ■
+■3a
b■+■
2b2
f
2c2 ■
-■5c
d■-■
3d 2
g2x
2 ■-■
xy■-
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h
4p2 ■
+■pq
■-■3
q2■
number AnD AlgebrA • pAtterns AnD AlgebrA
73Chapter 3 Algebra
10 mC ■a The■equivalent■of■(x■+■7)(x■-■2)■is:A x2■+■5x■-■14 B 2x■+■5 C x2■-■5x■-■14D x2■+■5x■+■14 E x2■-■5x■+■14
b What■is■the■equivalent■of■(4■-■y)(7■+■y)?A 28■-■y2 B 28■-■3y■+■y2 C 28■-■3y■-■y2
D 11■-■2y E 28■+■3y■-■y2
c The■equivalent■of■(2p■+■1)(p■-■5)■is:A 2p2■-■5 B 2p2■-■11p■-■5 C 2p2■-■9p■-■5D 2p2■-■6p■-5 E 2p2■+■9p -■5
unDerstAnDing
11 Expand■the■following■expressions■using■the■FOIL■method,■then■simplify.a (x■+■3)(x■-■3) b (x■+■5)(x■-■5) c (x■+■7)(x■-■7)d (x■-■1)(x■+■1) e (x■-■2)(x■+■2) f (2x■-■1)(2x■+■1)Can■you■see■a■pattern?■If■so,■explain.
12 Expand■the■following■expressions■using■the■FOIL■method,■then■simplify.a (x■+■1)(x■+■1) b (x■+■2)(x■+■2) c (x■+■8)(x■+■8)d (x■-■3)(x■-■3) e (x■-■5)(x■-■5) f (x■-■9)(x■-■9)Can■you■see■a■pattern?■If■so,■explain.
13 Simplify■the■following■expressions.a 2 1 3 4 7 3 1 1 4⋅ + ⋅ − ⋅ ⋅ +x x y y x y( ) ( )b ( )( )2 1 3 2 2 1 3 2⋅ − ⋅ ⋅ + ⋅x y x y ■c ( )3 4 5 1 2⋅ + ⋅x y
14 Two■rectangles■are■shown■at■right.The■difference■between■the■area■of■rectangle■A■and■rectangle■B■is■4x■+■8.■When■x■=■3,■the■ratio■of■the■area■of■rectangle■B■to■rectangle■A■is■1
2.■Find■the■
values■of■a■and■b.
15 For■the■box■shown■below■fi■nd■the■total■surface■area■and■the■volume■in■expanded■form.
4x + 3
3x - 1
x
reAsoning
16 For■each■of■the■following■shapes, i write■down■the■area■in■factor■formii expand■and■simplify■the■expressioniii discuss■any■limitations■on■the■value■of■x.
a (x + 2) m
(x - 1) m
b
(x + 5) cm
(2x - 1) cm
Rectangle A
Rec
tang
le
x + b
x + 2
x + 2
x + a
B
✔
✔
✔
x2■-■9 x2■-■25 x2■-■49x2■-■1 x2■-■4 4x2■-■1
x2■+■2x■+■1 x2■+■4x■+■4 x2■+■16x■+■64x2■-■6x■+■9 x2■-■10x■+■25 x2■-■18x■+■81
6.3x2■+■5.53xy■-■3.1y2
4.41x2■-■10.24y2
11.56x2■+■34.68xy■+■26.01y2
a■=■1■ b■=■5
16 a■ i■ (x■+■2)(x■-■1) ii x2■+■x■-■2 iii x■>■1
b i■ ( )( )( )2 1( ) 5( )5( )
2
x x( )x x( )( )x x( )( )2 1( )x x( )2 1( )− +( )− +( )( )− +( )( )2 1( )− +( )2 1( )x x− +x x( )x x( )− +( )x x( )( )x x( )− +( )x x( )( )2 1( )x x( )2 1( )− +( )2 1( )x x( )2 1( ) ii2 9 5
2
22 922 9x x2 9x x2 9+ −2 9+ −2 9x x+ −x x2 9x x2 9+ −2 9x x2 9iii x > 1
2
Surface■area■=■38x2■+■14x■-■6Volume■=■12x3■+■5x2■-■3x
number AnD AlgebrA • pAtterns AnD AlgebrA
74 maths Quest 9 for the Australian Curriculum
17 Show■that:(a■-■x)(a■+■x)■-■2(a■-■x)(a■-■x)■-■2x(a■-■x)■=■-(a■–■x)2
expansion patterns■■ Special■cases■when■expanding■brackets■will■be■examined■in■this■section.
Difference of two squares ■■ Difference of two squares■results■from■expanding■two■brackets■in■which■the■terms■are■identical■and■the■signs■are■opposite,■i.e.■+■and■-■.Consider■expanding■(x■+■3)(x■-■3).
■ (x■+■3)(x■-■3)■=■x(x■-■3)■+■3(x■-■3) =■x■ì■x■+■x■ì■-3■+■3■ì■x■+■3■ì■-3 =■x2■-■3x■+■3x■-■9 =■x2■-■9
■■ The■middle■terms,■-3x■+■3x,■cancel■each■other■out.■This■is■the■key■to■the■pattern■and■will■always■happen.
■■ Note:■The■terms■left■over■are■the■squares■of■each■of■the■original■terms.■In■other■words,■(x■+■3)(x■-■3)■=■x2■-■32.
■■ Notice■the■pattern■of■terms■in■the■pair■of■brackets■that■produce■the■difference■of■two■squares.Here■are■some■more■examples.
(x■+■5)(x■-■5)=■x2■-■52
=■x2■-■25
(x■+■4)(x■-■4)=■x2■-■42
=■x2■-■16
(x■+■h)(x■-■h)=■x2■-■h2
■
(2x■+■7)(2x■-■7)=■(2x)2■-■72
=■4x2■-■49■■ In■general,■(a + b)(a - b) = a2 - b2
Use the difference of two squares rule to expand and simplify each of the following.a (x + 8)(x - 8) b (6 - x)(6 + x)c (2x - 3)(2x + 3) d (3x + 5)(5 - 3x)
think Write
a 1 Write■the■expression. a (x■+■8)(x■-■8)
2 Check■that■the■expression■can■be■written■as■the■difference■of■two■squares■by■comparing■it■with■(a■+■b)(a■-■b).■It■can.
3 Write■the■answer■as■the■difference■of■two■squares■using■the■formula■(a■+■b)(a■-■b)■=■a2■-■b2,■where■a■=■x■and■b■=■8.
=■x2■-■82
=■x2■-■64
b 1 Write■the■expression. b (6■-■x)(6■+■x)
2 Check■that■the■difference■of■two■squares■rule■can■be■used.■It■can.
Note:■(6■-■x)(6■+■x)■is■the■same■as■(6■+■x)(6■-■x).
3 Write■the■answer■as■the■difference■of■two■squares■using■the■formula
=■62■-■x2
=■36■-■x2
(a■+■b)(a■-■b)■=■a2■-■b2,■where■a■=■6■and■b■=■x.
refleCtion
Explain why, when expanded, (x + y )(2x + y ) gives the same result as (2x + y )(x + y ).
3e
WorkeD exAmple 15
Answers■may■vary.
number AnD AlgebrA • pAtterns AnD AlgebrA
75Chapter 3 Algebra
c 1 Write■the■expression. c (2x■-■3)(2x■+■3)
2 Check■that■the■difference■of■two■squares■rule■can■be■used.■It■can.
3 Write■the■answer■as■the■difference■of■two■squares■using■the■formula
=■(2x)2■-■32
=■4x2■-■9
(a■+■b)(a■-■b)■=■a2■-■b2,■where■a■=■2x■and■b■=■3.
d 1 Write■the■expression. d (3x + 5)(5■-■3x)
2 Check■that■the■difference■of■two■squares■rule■can■be■used■by■rearranging■the■terms.
(5■+■3x)(5■-■3x)
3 Write■the■answer■as■the■difference■of■two■squares■using■the■formula■(a■+■b)(a■-■b)■=■a2■-■b2,■where■a■=■5■and■b■=■3x.
=■52■-■(3x)2
=■25■-■9x2
the expansion of a perfect square■■ A■perfect square■is■a■number■multiplied■by■itself.■For■example,■1,■4,■9,■16■…■and■so■on■are■all■perfect■squares■since■1■=■1■ì■1■=■12,■4■=■2■ì■2■=■22,■9■=■3■ì■3■=■32,■16■=■4■ì■4■=■42■and■so■on.■
■■ Similarly,■(x■+■3)2■is■a■perfect■square■since■it■is■equivalent■to■(x■+■3)(x■+■3).■When■expanding■a■perfect■square,■the■following■pattern■can■be■seen.■
■■ Consider:(x■+■3)(x■+■3)■=■x(x■+■3)■+■3(x■+■3)
=■x■ì■x■+■x■ì■3■+■3■ì■x■+■3■ì■3 =■x2■+■3x■+■3x■+■9 =■x2■+■6x■+■9
■■ Use■the■FOIL■method:(x■+■3)2■=■(x■+■3)(x■+■3)
=■x2■+■3x■+■3x■+■9=■x2■+■6x■+■9
■■ In■general:(a■+■b)2■=■(a■+■b)(a■+■b)
=■a2■+■ab +■ba +■b2■ using■the■FOIL■method =■a2■+■2ab +■b2
(a■-■b)2■=■(a■-■b)(a■-■b) =■a2■-■ab -■ba +■b2■ using■the■FOIL■method =■a2■-■2ab +■b2
■■ This■pattern■can■also■be■described■in■words:■Square■the■first■term,■add■the■square■of■the■last■term■and■then■add■(or■subtract)■twice■their■product.
Use the perfect squares technique to expand and simplify the following.a (x + 1)(x + 1) b (x - 2)2 c (2x + 5)2 d (4x - 5y)2
think Write
a 1 Write■the■expression. a (x■+■1)(x■+■1)
2 Apply■the■formula■for■perfect■squares:■(a■+■b)(a■+■b)■=■a2■+■2ab■+■b2,■where■a■=■x■and■b■=■1.
=■(x)2■+■2■ì■x■ì■1■+■(1)2
3 Simplify. =■x2■+■2x■+■1
WorkeD exAmple 16
number AnD AlgebrA • pAtterns AnD AlgebrA
76 maths Quest 9 for the Australian Curriculum
b 1 Write■the■expression. b (x■-■2)2
=■(x■-■2)(x■-■2)
2 Apply■the■formula■for■perfect■squares:■(a■-■b)(a■-■b)■=■a2■-■2ab■+■b2,■where■a■=■x■and■b■=■2.
=■(x)2■-■2■ì■x■ì■2■+■(2)2
3 Simplify. =■x2■-■4x■+■4
c 1 Write■the■expression. c (2x■+■5)2
=■(2x■+■5)(2x■+■5)
2 Apply■the■formula■for■perfect■squares:■(a■+■b)(a■+■b)■=■a2■+■2ab■+■b2,■where■a■=■2x■and■b■=■5.
=■(2x)2■+■2■ì■2x■ì■5■+■(5)2
3 Simplify. =■4x2■+■20x■+■25
d 1 Write■the■expression. d (4x■-■5y)2
=■(4x■-■5y)(4x■-■5y)
2 Apply■the■formula■for■perfect■squares:■(a■-■b)(a■-■b)■=■a2■-■2ab■+■b2,■where■a■=■4x■and■b■=■5y.
=■(4x)2■-■2■ì■4x■ì■5y■+■(5y)2
3 Simplify. =■16x2■-■40xy■+■25y2
remember
1.■ The■difference■of■two■squares■rule■is:■(a■+■b)(a■-■b)■=■a2■-■b2.2.■ The■expansion■of■a■perfect■square:■The■identical■brackets■(perfect■squares)■rules■are:
(a■+■b)(a■+■b)■=■a2■+■2ab■+■b2
(a■-■b)(a■-■b)■=■a2■-■2ab■+■b2.
expansion patternsfluenCY
1 We 15a, b Use■the■difference■of■two■squares■rule,■to■expand■and■simplify■each■of■the■following.a (x■+■2)(x■-■2) b (y■+■3)(y■-■3)c (m■+■5)(m■-■5) d (a■+■7)(a■-■7)e (x■+■6)(x■-■6) f (p■-■12)(p■+■12)g (a■+■10)(a■-■10) h (m■-■11)(m■+■11)
2 We 15c, d Use■the■difference■of■two■squares■rule■to■expand■and■simplify■each■of■the■following.a (2x■+■3)(2x■-■3) b (3y■-■1)(3y■+■1)c (5d■-■2)(5d■+■2) d (7c■+■3)(7c■-■3)e (2■+■3p)(2■-■3p) f (1■-■9x)(1■+■9x)g (5■-■12a)(5■+■12a) h (3■+■10y)(3■-■10y)i (2b■-■5c)(2b■+■5c) j (10-2x)(2x+10)
3 We 16a, b Use■the■perfect■squares■rule■to■expand■and■simplify■each■of■the■following.a (x■+■2)(x■+■2) b (a■+■3)(a■+■3)c (b■+■7)(b■+■7) d (c■+■9)(c■+■9)e (m■+■12)2 f (n■+■10)2
g (x■-■6)2 h (y■-■5)2
i (9■-■c)2 j (8■+■e)2
k 2(x■+■y)2 l (u■-■v)2
exerCise
3e
eBookpluseBookplus
Activity 3-E-1Exploring expansion
patternsdoc-3990
Activity 3-E-2Using expansion
patternsdoc-3991
Activity 3-E-3Recognising
expansion patternsdoc-3992
inDiViDuAl pAthWAYs
x2■-■4 y2■-■9m2■-■25 a2■-■49
x2■-■36 p2■-■144a2■-■100 m2■-■121
4x2■-■9 9y2■-■125d 2■-■4 49c2■-■94■-■9p2 1■-■81x2
25■-■144a2 9■-■100y2
4b2■-■25c2 100■-■4x2
x2■+■4x■+■4 a2■+■6a■+■9b2■+■14b■+■49 c2■+■18c■+■81
m2■+■24m■+■144 n2■+■20n■+■100x2■-■12x■+■36 y2■-■10y■+■2581■-■18c■+■c2 64■+■16e■+■e2
2x2■+■4xy■+■2y2 u2■-■2uv■+■v2
number AnD AlgebrA • pAtterns AnD AlgebrA
77Chapter 3 Algebra
4 We 16c, d Use■the■perfect■squares■rule■to■expand■and■simplify■each■of■the■following.a (2a■+■3)2 b (3x■+■1)2
c (2m■-■5)2 d (4x■-■3)2
e (5a■-■1)2 f (7p■+■4)2
g (9x■+■2)2 h (4c■-■6)2
i (3■+■2a)2 j (5■+■3p)2
k (2■-■5x)2 l (7■-■3a)2
m (9x■-■4y)2 n (8x■-■3y)2
5 Expand■and■simplify■the■following■expressions.a (2.2x■+■4y)2 b (3.2x■-■4.5y)2
unDerstAnDing
6 Francis■has■fenced■off■an■area■in■her■paddock■for■spring■lambs.■The■area■of■the■paddock■is■9x2■+■6x■+■1■m2.■By■using■pattern■recognition,■fi■nd■the■side■length,■in■terms■of■x,■of■the■paddock.
reAsoning
7 A■square■of■side■length■x■cm■is■drawn.■a For■the■square,■write■down■an■expression■for■its: i perimeter,■in■cm■ ii area,■in■cm2.■b A■1-cm■strip■is■removed■from■one■side■and■added■to■the■adjacent■side■to■form■another■
plane■fi■gure. i Determine■the■perimeter■of■the■new■shape. ii Determine■the■area,■in■cm2,■of■the■new■shape.c Explain■why■the■perimeter■changes■but■the■areas■remain■the■same.
8x
x
y
y
a What■is■the■perimeter■of■this■fi■gure?b What■is■the■area?■(Express■as■an■answer■and■as■the■product■of■the■lengths■of■the■sides.)■
What■have■you■generalised?
more complicated expansionsexpanding more than two brackets
■■ It■is■possible■to■expand■more■than■two■brackets,■such■as■expanding■three■brackets,■four■brackets,■and■so■on.
refleCtion
How could you represent (x - 3)2 on a diagram?
3feBookpluseBookplus
InteractivityExpanding
bracketsint-2763
4a2■+■12a■+■9 9x2■+■6x■+■1
4m2■-■20m■+■25 16x2■-■24x■+■9
25a2■-■10a■+■1 49p2■+■56p■+■16
81x2■+■36x■+■4 16c2■-■48c■+■36
9■+■12a■+■4a2 25■+■30p■+■9p2
4■-■20x■+■25x2 49■-■42a■+■9a2
81x2■-■72xy■+■16y2 64x2■-■48xy■+■9y2
4.84x2■+■17.6xy■+■16y2
10.24x2■-■28.8xy■+■20.25y2
(3x +■1)m
4x
x2
4x + 2
x2
Answers■will■vary.
4y
y2■-■x2■=■(y■+■x)(y■-■x)■ Difference■of■two■squares.
number AnD AlgebrA • pAtterns AnD AlgebrA
78 maths Quest 9 for the Australian Curriculum
Expand and simplify each of the following expressions.a (x + 3)(x + 4) + 4(x - 2) b (x - 2)(x + 3) - (x - 1)(x + 2) c (x + 2)(x - 2) - (x + 2)(x + 2) d 2(x + 3)(x - 4) + (x - 2)2
think Write
a 1 Write■the■expression. a (x■+■3)(x■+■4)■+■4(x■-■2)
2 Expand■and■simplify■the■first■pair■of■brackets.
=■x2■+■4x■+■3x■+■12■+■4(x■-■2)
3 Expand■the■last■bracket. =■x2■+■4x■+■3x■+■12■+■4x■-■8
4 Simplify■by■collecting■like■terms. =■x2■+■11x■+■4
b 1 Write■the■expression. b (x■-■2)(x■+■3)■-■(x■-■1)(x■+■2)
2 Expand■and■simplify■the■first■pair■of■brackets.
=■x2■+■3x■-■2x■-■6■-■(x■-■1)(x■+■2)=■x2■+■x■-■6■-■(x■-■1)(x■+■2)■
3 Expand■and■simplify■the■second■pair■of■brackets.■Take■care■to■keep■the■expanded■form■of■the■second■pair■of■factors■in■a■bracket.
=■x2■+■x■-■6■-■(x2■+■2x■-■1x■-■2)
4 Subtract■all■of■the■second■result■from■the■first■result.■Remember■that■■-(x2■+■x■-■2)■=-1(x2■+■x■-■2).
=■x2■+■x■-■6■-■(x2■+■x■-■2)=■x2■+■x■-■6■-■x2■-■x■+■2
5 Simplify■by■collecting■like■terms. =-4
c 1 Write■the■expression. c (x■+■2)(x■-■2)■-■(x■+■2)(x■+■2)
2 Expand■and■simplify■the■first■pair■of■brackets.■It■is■a■difference■of■two■squares■expansion.
=■x2■-■22■-■(x■+■2)(x■+■2)=■x2■-■4■-■(x■+■2)(x■+■2)
3 Expand■and■simplify■the■second■pair■of■brackets.■It■is■a■perfect■square■expansion■(that■is,■an■identical■bracket■expansion.)■Remember■to■keep■the■second■expansion■in■brackets.
=x2■-■4■-■(x2■+■2■ì■x■ì■2■+■22)=■x2■-■4■-■(x2■+■4x■+■4)
4 Subtract■all■of■the■second■result■from■the■first■result.
=■x2■-■4■-■x2■-■4x■-■4
5 Simplify■by■collecting■like■terms. =■-4x■-■8
d 1 Write■the■expression. d 2(x■+■3)(x■-■4)■+■(x■-■2)2
=■2(x■+■3)(x■-■4)■+■(x■-■2)(x■-2)
2 Expand■the■first■pair■of■brackets,■and■then■multiply■by■the■coefficient■of■2■outside■the■pair.
=■2(x2■-■4x■+■3x■-■12)■+■(x■-■2)(x■-2)=■2(x2■-■x■-■12)■+■(x■-■2)(x■-2)=2x2■-■2x■-■24■+■(x■-■2)(x■-2)
3 Expand■the■second■pair■of■brackets.■It■is■a■perfect■square■expansion■(an■identical■bracket■expansion).
=2(x2■-■4x■+■3x■-■12)■+■(x2■-■2■ì■x■ì■2■+■22)
4 Add■the■two■results. =■2x2■-■2x■-■24■+■x2■-■4x■+■4
5 Simplify■by■collecting■like■terms. =■3x2■-■6x■-■20
WorkeD exAmple 17
number AnD AlgebrA • pAtterns AnD AlgebrA
79Chapter 3 Algebra
remember
1.■ Brackets■or■pairs■of■brackets■that■are■added■or■subtracted■must■be■expanded■separately.
2.■ Read■the■mathematics■slowly■and■identify■if■there■is■an■expansion■pattern■that■can■be■applied■directly.■That■is,■can■the■difference■of■perfect■squares■or■the■expansion■of■perfect■squares■be■applied?
3.■ Always■collect■any■like■terms■following■an■expansion.■Take■care■when■the■second■pair■of■brackets■is■subtracted■from■the■fi■rst■pair;■use■of■another■bracket■is■recommended.
more complicated expansionsfluenCY
We17 Expand■and■simplify■each■of■the■following■expressions. 1 (x■+■3)(x■+■5)■+■(x■+■2)(x■+■3) 2 (x■+■4)(x■+■2)■+■(x■+■3)(x■+■4) 3 (x■+■5)(x■+■4)■+■(x■+■3)(x■+■2) 4 (x■+■1)(x■+■3)■+■(x■+■2)(x■+■4) 5 (p■-■3)(p■+■5)■+■(p■+■1)(p■-■6) 6 (a■+■4)(a■-■2)■+■(a■-■3)(a■-■4) 7 (p■-■2)(p■+■2)■+■(p■+■4)(p■-■5) 8 (x■-■4)(x■+■4)■+■(x■-■1)(x■+■20) 9 (y■-■1)(y■+■3)■+■(y■-■2)(y■+■2) 10 (d■+■7)(d■+■1)■+■(d■+■3)(d■-■3)11 (x■+■2)(x■+■3)■+■(x■-■4)(x■-■1) 12 (y■+■6)(y■-■1)■+■(y■-■2)(y■-■3)13 (x■+■2)2■+■(x■-■5)(x■-■3) 14 (y■-■1)2■+■(y■+■2)(y■-■4)15 (p■+■2)(p■+■7)■+■(p■-■3)2 16 (m■-■6)(m■-■1)■+■(m■+■5)2
17 (x■+■3)(x■+■5)■-■(x■+■2)(x■+■5) 18 (x■+■5)(x■+■2)■-■(x■+■1)(x■+■2)19 (x■+■3)(x■+■2)■-■(x■+■4)(x■+■3) 20 (m■-■2)(m■+■3)■-■(m■+■2)(m■-■4)21 (b■+■4)(b■-■6)■-■(b■-■1)(b■+■2) 22 (y■-■2)(y■-■5)■-■(y■+■2)(y■+■6)23 (p■-■1)(p■+■4)■-■(p■-■2)(p■-■3) 24 (x■+■7)(x■+■2)■-■(x■-■3)(x■-■4)25 (c■-■2)(c■-■1)■-■(c■+■6)(c■+■7) 26 (f■-■7)(f■+■2)■-■(f■+■4)(f■+■5)27 (m■+■3)2■-■(m■+■4)(m■-■2) 28 (a■-■6)2■-■(a■-■2)(a■-■3)29 (p■-■3)(p■+■1)■-■(p■+■2)2 30 (x■+■5)(x■-■4)■-■(x■-■1)2
the highest common factor■■ The■term■expanding■is■defi■ned■as■changing■a■compact■form■of■an■expression■that■is■in■terms■of■factors■to■an■expanded■form.■
■■ Factorising■is■the■reverse■operation■of■expansion.■■■ Factorising■an■expression■transforms■the■expression■to■a■more■compact■form■in■which■it■is■written■as■a■product■of■factors.■For■example:■ 12■=■3■ì4■ (factorised■form)
=■4(2■+■1)■ (factorised■form)■ =■4■ì■2■+■4■ì■1■ (expanded■form)■
■■ To■factorise,■all■factors■of■the■integers■need■to■be■known.
factors■■ The■factors■of■an■integer■are■two■or■more■integers■that,■when■multiplied■together,■produce■that■integer.■
exerCise
3f
eBookpluseBookplus
Activity 3-F-1Reviewing expansion
methodsdoc-3993
Activity 3-F-2Applying expansion
methodsdoc-3994
Activity 3-F-3Complex algebraic
expansionsdoc-3995
inDiViDuAl pAthWAYs
eBookpluseBookplus
Digital docsSkillSHEET 3.20
doc-6143
SkillSHEET 3.21doc-6144
refleCtion
On a diagram how would you show (m - 2)(m + 3) - (m + 2)(m - 4)?
3g
2x2■+■13x■+■21 2x2■+■13x■+■202x2■+■14x■+■26 2x2■+■10x■+■112p2■-■3p■-■21 2a2■-■5a■+■42p2■-■p■-■24 2x2■+■19x■-■362y2■+■2y■-■7 2d2■+■8d■-■22x2■+■10 2y2
2x2■-■4x■+■19 2y2■-■4y■-■72p2■+■3p■+■23 2m2■+■3m■+■31
x■+■5 4x■+■8-2x■-■6 3m■+■2-3b■-■228p■-■10-16c■-■40
4m■+■17-6p■-■7
22 -15y■-■224 16x■+■226 -14f■-■3428 -7a■+■3030 3x■-■21
number AnD AlgebrA • pAtterns AnD AlgebrA
80 maths Quest 9 for the Australian Curriculum
For■example:■ 3■ì■2■=■6,■so■3■and■2■are■factors■of■6.■■ A■factor■of■a■number■is■an■integer■such■that■when■the■factor■is■divided■into■the■number■there■is■no■remainder.■
■■ Factor■pairs■of■a■term■are■numbers■and/or■pronumerals■that,■when■multiplied■together,■produce■the■original■term.
Find all the factors of 12 and list them in ascending order.
think Write
1 List■pairs■of■integers■which■when■multiplied■produce■12.Note:■5■is■not■a■factor■of■12■because■125
■=■225.■It■does■not■divide■exactly■into■12.
1,■12;■2,■6;■3,■4
2 List■the■factors■of■12■in■ascending■order.
Factors■of■12■are■1,■2,■3,■4,■6,■12.
finding the highest Common factor (hCf)■■ The■highest common factor■or■HCF■of■two■or■more■numbers■is■the■largest■factor■that■divides■into■all■of■the■given■numbers■without■a■remainder.■This■also■applies■to■algebraic■terms.
■■ The■highest■common■factor■of■xyz■and■2yz■is■yz■because:
xyz■=■x■ì■y■ì■z■ 2yz■=■2■ì■y■ì■z.
■■ The■HCF■is■yz■(combining■the■common■factors■of■each).■■ For■an■algebraic■term,■the■highest■common■factor■is■found■by■taking■the■HCF■of■the■coeffi■cients■and■combining■all■common■pronumerals.
Find the highest common factor (HCF) of each of the following.a 12, 16 and 56b 4abc and 6bcd
think Write
a 1 Find■the■factors■of■12■and■write■them■in■ascending■order.
a 12:■1,■12;■2,■6;■3,■4Factors■of■12■are■1,■2,■3,■4,■6,■12.
2 Find■the■factors■of■16■and■write■them■in■ascending■order.
16:■1,■16;■2,■8;■4,■4Factors■of■16■are■1,■2,■4,■8,■16.
3 Find■the■factors■of■56■and■write■them■in■ascending■order.
56:■1,■56;■2,■28;■4,■14;■7,■8Factors■of■56■are■1,■2,■4,■7,■8,■14,■28,■56.
4 Write■the■common■factors. Common■factors■are■1,■2,■4.
5 Find■the■highest■common■factor■(HCF). The■HCF■of■12,■16■and■56■is■4.
WorkeD exAmple 18
WorkeD exAmple 19
number AnD AlgebrA • pAtterns AnD AlgebrA
81Chapter 3 Algebra
b 1 Find■the■factors■of■4. b 4:■1,■2,■4
2 Find■the■factors■of■6. 6:■1,■2,■3,■6
3 Write■the■common■factors■of■4■and■6. Common■factors■of■4■and■6■are■1,■2.
4 Find■the■highest■common■factor■of■the■coefficients.
The■HCF■of■4■and■6■is■2.
5 List■the■pronumerals■that■are■common■to■each■term.
Common■pronumerals■are■b■and■c.
6 Find■the■HCF■of■the■algebraic■terms■by■multiplying■the■HCF■of■the■coefficients■to■all■the■common■pronumerals.
The■HCF■of■4abc■and■6bcd■is■2bc.
factorising expressions by finding the highest common factor
■■ An■algebraic■expression■is■made■up■of■terms■that■are■separated■by■either■a■+■or■a■-■sign.■For■example:
■ 4xy■+■12x■-■4xy2■is■an■algebraic■expression■that■has■3■terms.■ 4xy,■12x■and■4xy2■are■all■terms.
■■ To■factorise■such■an■expression,■find■the■highest■common■factor■of■the■terms.■In■this■expression,■4x■is■the■highest■common■factor.■
■ 4xy■+■12x■-■4xy2■ =■4xy■+■3■ì■4x■–■4xy2■ =■4x(y■+■3■–■y2)
■■ As■can■be■seen■above,■each■term■in■the■expression■is■written■as■a■product■of■two■factors,■one■being■the■HCF.■
■■ The■HCF■is■placed■outside■the■brackets■and■the■remaining■terms■are■placed■inside■the■brackets.■■■ To■check■is■the■factorisation■is■correct,■expand■the■brackets.■■ The■expanded■form■should■be■the■original■expression,■if■the■factorisation■is■correct.■
Factorise each of the following expressions by first finding the highest common factor (HCF).a 5x + 15y b -14xy - 7yc 15ab - 21bc + 18bf d 6x2y + 9xy2
think Write
a 1 Find■the■HCF■of■the■coefficients.■List■the■pronumerals■common■to■each■term.
a The■HCF■of■5■and■15■is■5.There■are■no■common■pronumerals.■Therefore■the■highest■common■factor■of■the■expression■is■5.
2 Write■the■expression. 5x■+■15y3 Write■each■term■in■the■expression■as■
a■product■of■two■factors,■one■being■the■HCF.
=■5■ì■x■+■5■ì■3y
4 Factorise■the■expression■by■placing■the■HCF■outside■the■brackets■and■the■remaining■terms■inside■the■brackets.■■An■expansion■of■the■brackets■■should■return■you■to■the■original■expression.■
=■5(x■+■3y)
WorkeD exAmple 20
number AnD AlgebrA • pAtterns AnD AlgebrA
82 maths Quest 9 for the Australian Curriculum
b 1 Find■the■HCF■of■14■and■7.■List■the■pronumerals■common■to■each■term.
b The■HCF■of■14■and■7■is■7.The■common■pronumeral■is■y.Therefore■7y■is■common■to■both■terms.
2 Write■the■expression. -14xy■-■7y
3 Write■each■term■in■the■expression■as■a■product■of■two■factors,■one■being■the■HCF.
=■7y■ì■-2x■-■7y■ì■1
4 It■can■also■be■seen■that■-1■is■common■to■both■terms■as■well■■as■7.
5 Place■the■-7y■outside■the■brackets■and■the■remaining■■terms■inside■the■brackets.■An■expansion■of■the■brackets■will■result■in■the■original■expression■indicating■that■the■factorisation■is■correct.
=■-7y(2x■+■1)
c 1 Find■the■HCF■of■the■coefficients,■using■only■positive■integer■factors.■List■the■pronumerals■common■to■each■term.
c The■HCF■of■15,■21■and■18■is■3.The■common■pronumeral■is■b.Therefore■the■highest■common■factor■of■the■expression■is■3b.
2 Write■the■expression. 15ab■-■21bc■+■18bf
3 Write■each■term■in■the■expression■as■a■product■of■two■factors,■one■being■the■HCF.
=■3b■ì■5a■-■3b■ì■7c■+■3b■ì■6f
4 Place■the■HCF■outside■the■brackets■and■the■remaining■terms■inside■the■brackets.■Expand■your■result■to■check■that■your■factorisation■is■correct.
=■3b(5a■-■7c■+■6f)
d 1 Find■the■HCF■of■the■coefficients.■List■the■pronumerals■common■to■each■term.
d The■HCF■of■6■and■9■is■3.The■common■pronumerals■are■x■and■y.Therefore■the■highest■common■factor■of■the■expression■is■3xy.
2 Write■the■expression. 6x2y■+■9xy2
3 Write■each■term■in■the■expression■as■a■product■of■two■factors,■one■being■the■HCF.
=■3xy■ì■2x■+■3xy■ì■3y
4 Place■the■HCF■outside■the■brackets■and■the■remaining■terms■inside■the■brackets.■Expand■your■result■to■check■that■the■factorisation■is■correct.
=■3xy(2x■+■3y)
number AnD AlgebrA • pAtterns AnD AlgebrA
83Chapter 3 Algebra
remember
1.■ Factorising■is■the■opposite■of■expanding.■Factorising■is■the■process■that■transforms■an■expanded■form■to■a■more■compact■form■that■consists■of■two■or■more■factors■multiplied■together.
2.■ Factor■pairs■of■a■term■are■numbers■and/or■pronumerals■that,■when■multiplied■together,■produce■the■original■term.
3.■ The■number■itself■and■1■are■factors■of■every■integer.4.■ The■highest■common■factor■(HCF)■of■given■terms■is■the■largest■factor■that■divides■into■
all■terms■without■a■remainder.5.■ An■expression■is■factorised■by:
(a)■ fi■nding■the■HCF■of■the■terms(b)■writing■each■term■in■the■expression■as■a■product■of■two■factors,■one■being■the■HCF(c)■ placing■the■HCF■outside■the■brackets■and■the■remaining■terms■inside■the■brackets.■
Pay■particular■note■to■the■signs■of■the■terms.6.■ Always■check■that■your■factorisation■is■correct■by■doing■a■quick■expansion■of■the■
brackets,■which■should■result■in■your■original■expression.
the highest common factorfluenCY
1 We 18 Find■all■the■factors■of■each■of■the■following■integers.a 36 b 17 c 51 d -14 e -8 f 100g -42 h 32 i -32 j -9 k -64 l -81m 29 n -92 o 48 p -12
2 We 19 Find■the■highest■common■factor■(HCF)■of■each■of■the■following.a 4■and■12 b 6■and■15 c 10■and■25d 24■and■32 e 12,■15■and■21 f 25,■50■and■200g 17■and■23 h 6a■and■12ab i 14xy■and■21xzj 60pq■and■30q k 50cde■and■70fgh l 6x2■and■15xm 6a■and■9c n 5ab■and■25 o 3x2y■and■4x2zp 4k■and■6
3 mC ■What■is■5m■the■highest■common■factor■of?A 2m■and■5m B 5m■and■m C 25mn■and■15lmD 20m■and■40m E 15m2n■and■5n2
4 We20 Factorise■each■of■the■following■expressions.a 4x■+■12y b 5m■+■15n c 7a■+■14bd 7m■-■21n e -8a■-■24b f 8x■-■4yg -12p■-■2q h 6p■+■12pq■+■18q i 32x■+■8y■+■16zj 16m■-■4n■+■24p k 72x■-■8y■+■64pq l 15x2■-■3ym 5p2■-■20q n 5x■+■5 o 56q■+■8p2
p 7p■-■42x2y q 16p2■+■20q■+■4 r 12■+■36a2b■-■24b2
5 Factorise■each■expression.a 9a■+■21b b 4c■+■18d2 c 12p2■+■20q2
d 35■-■14m2n e 25y2■-■15x f 16a2■+■20bg 42m2■+■12n h 63p2■+■81■-■27y i 121a2■-■55b■+■110cj 10■-■22x2y3■+■14xy k 18a2bc■-■27ab■-■90c l 144p■+■36q2■-■84pqm 63a2b2■-■49■+■56ab2 n 22■+■99p3q2■-■44p2r o 36■-■24ab2■+■18b2c
exerCise
3g
eBookpluseBookplus
Activity 3-G-1Reviewing HCF in
factorisationdoc-3996
Activity 3-G-2Using HCF in factorisation
doc-3997
Activity 3-G-3Applying HCF in
factorisationdoc-3998
inDiViDuAl pAthWAYs
eBookpluseBookplus
Digital docSkillSHEET 3.9
doc-6130
eBookpluseBookplus
Digital docSkillSHEET 3.10
doc-6131
1
a■■-3
6,■-
18,■-
12,■-
9,■-
6,■-
4,■-
3,■-
2,■-
1,■1
,■2,■3
,■4,■6
,■9,■1
2,■1
8,■3
6
b
-17,
■-1,
■1,■1
7
c
-51,
■-17
,■-3,
■-1,
■1,■3
,■17,
■51
d-1
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,■2,■7
,■14
e-8
,■-4,
■-2,
■-1,
■1,■2
,■4,■8
f-1
00,■-
50,■-
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20,■-
10,■-
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,■2,■4
,■5,■1
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00g
-4
2,■-
21,■-
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7,■-
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,■6,■7
,■14,
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■42
4 3 58 3 251 6a 7x
30q 10 3x3 5 x2
2
✔
4(x■+■3y) 5(m■+■3n) 7(a■+■2b)7(m■-■3n) -8(a■+■3b) 4(2x■-■y)-2(6p■+■q)
3(3a■+■7b) 2(2c■+■9d2) 4(3p2■+■5q2)7(5■-■2m2n) 5(5y2■-■3x) 4(4a2■+■5b)6(7m2■+■2n)
4 h■ 6(p■+■2pq■+■3q) i 8(4x■+■y■+■2z)
j 4(4m■-■n■+■6p) k 8(9x■-■y■+■8pq)
l 3(5x2■-■y) m 5(p2■-■4q)
n 5(x■+■1) o 8(7q■+■p2)
p 7(p■-■6x2y) q 4(4p2■+■5q■+■1)
r 12(1■+■3a2b■-■2b2)
5 h■ 9(7p2■+■9■-■3y) i 11(11a2■-■5b■+■10c) j 2(5■-■11x2y3■+■7xy) k 9(2a2bc■-■3ab■-■10c) l 12(12p■+■3q2■-■7pq) 6(6■-■4ab2■+■3b2c)11(2■+■9p3q2■-■4p2r)7(9a2b2■-■7■+■8ab2)
number AnD AlgebrA • pAtterns AnD AlgebrA
84 maths Quest 9 for the Australian Curriculum
6 Factorise■the■following■expressions.a -x■+■5 b -a■+■7 c -b■+■9d -2m■-■6 e -6p■-■12 f -4a■-■8g -3n2■+■15m h -7x2y2■+■21 i -7y2■-■49zj -12p2■-■18q k -63m■+■56 l -12m3■-■50x3■m -9a2b■+■30 n -15p■-■12q o -18x2■+■4y2
p -3ab■+■18m■-■21 q -10■-■25p2■-■45q r -90m2■+■27n■+■54p3
7 Factorise■each■of■the■following■expressions.■a a2■+■5a b m2■+■3m c x2■-■6xd 14q■-■q2 e 18m■+■5m2 f 6p■+■7p2
g 7n2■-■2n h a2■-■ab■+■5a i 7p■-■p2q■+■pqj xy■+■9y■-■3y2 k 5c■+■3c2d■-■cd l 3ab■+■a2b■+■4ab2
m 2x2y■+■xy■+■5xy2 n 5p2q2■-■4pq■+■3p2q o 6x2y2■-■5xy■+■x2y
8 Factorise■each■of■the■following■expressions.a 5x2■+■15x b 10y2■+■2y c 12p2■+■4pd 24m2■-■6m e 32a2■-■4a f -2m2■+■8mg -5x2■+■25x h -7y2■+■14y i -3a2■+■9aj -12p2■-■2p k -15b2■-■5b l -26y2■-■13ym 4m■-■18m2 n -6t■+■36t2 o -8p■-■24p2
unDerstAnDing
9 A■large■billboard■display■is■in■the■shape■of■a■rectangle■as■shown■at■right.■There■are■3■regions■(A,■B,■C)■with■dimensions■in■terms■of■x■as■shown.a Determine■the■total■area■of■the■rectangle.■Give■your■
answer■in■factorised■form.b Determine■the■area■of■each■region■in■the■simplest■form■as■
possible.
reAsoning
10 On■her■recent■Algebra■test,■Marcia■wrote■down■her■answer■of■4ab(a■+■1)■to■the■question■‘Using■factorisation,■simplify■the■following■expression■(a■+■1)(a■+■b)2■-■(a■+■1)(a■-■b)2■’.■If■Marcia■used■difference■of■two■squares■in■her■solution,■explain■the■steps■she■took■to■get■her■answer.■
more factorising using the highest common factor
■■ When■factorising,■always■look■for■highest■common■factors■first.
the binomial common factor■■ Common■factors■can■be■expressions.■■ Consider■the■following■expression:■5(x■+■y)■+■6b(x■+■y).
(x + 3 )
(x + 1 )
x
2x
B
A
C
refleCtion
How do you find the factors of terms within algebraic expressions?
3h
-(x■-■5) -(a■-■7) -(b■-■9)
-2(m■+■3) -6(p■+■2) -4(a■+■2)
-3(n2■-■5m) -7(x2y2■-■3) -7(y2■+■7z)-6(2p2■+■3q) -7(9m■-■8) -2(6m3■+■25x3)
-3(3a2b■-■10) -3(5p■+■4q) -2(9x2■-■2y2)
-5(2■+■5p2■+■9q) -9(10m2■-■3n■-■6p3)
a(a■+■5) m(m■+■3) x(x■-■6)q(14■-■q) m(18■+■5m) p(6■+■7p)
n(7n■-■2) a(a■-■b■+■5) p(7■-■pq■+■q)
y(x■+■9■-■3y) c(5■+■3cd■-■d) ab(3■+■a■+■4b)xy(2x■+■1■+■5y)
pq(5pq■-■4■+■3p)xy(6xy■-■5■+■x)
5x(x■+■3) 2y(5y■+■1) 4p(3p■+■1)
6m(4m■-■1) 4a(8a■-■1) -2m(m■-■4)-5x(x■-■5) -7y(y■-■2) -3a(a■-■3)-2p(6p■+■1) -5b(3b■+■1) -13y(2y■+■1)2m(2■-■9m) -6t(1■-■6t) -8p(1■+■3p)
2(x■+■3)(4x■+■1)
A■=■x(x■+■3)■ C■=■2(x■+■3)x■ B■=■5x2■+■17x■+■6
Answers■will■vary.
-3(ab■-■6m■+■7)
number AnD AlgebrA • pAtterns AnD AlgebrA
85Chapter 3 Algebra
■■ Both■terms■contain■the■bracketed■expression■(x■+■y).■■■ Therefore■(x■+■y)■is■a■common■factor■of■both■terms.■■■ This■is■called■a■binomial factor■because■it■is■an■expression■that■contains■two■terms.■
Factorise each of the following expressions.a 5(x + y) + 6b(x + y)b 2b(a - 3b) - (a - 3b)In both these expressions it can be seen that the HCF is a binomial factor.
think Write
a 1 Identify■the■common■factor. a The■common■factor■is■(x■+■y).
2 Write■the■expression. 5(x■+■y)■+■6b(x■+■y)
3 Write■each■term■in■the■expression■as■a■product■of■two■factors,■one■being■the■HCF.
=■5■ì■(x■+■y)■+■6b■ì■(x■+■y)
4 Factorise■by■taking■out■the■binomial■common■■factor■and■placing■the■remaining■terms■inside■brackets.
=■(x■+■y)(5■+■6b)
b 1 Identify■the■common■factor. b The■common■factor■is■(a■-■3b).
2 Write■the■expression. 2b(a■-■3b)■-■(a■-■3b)
3 Write■each■term■in■the■expression■as■a■product■of■two■factors,■one■being■the■HCF.
=■2b■ì■(a■-■3b)■-■1(a■-■3b)
4 Factorise■by■taking■out■the■binomial■common■■factor■and■placing■the■remaining■terms■inside■brackets.
=■(a■-■3b)(2b■-■1)
■■ These■answers■can■also■be■checked■by■expanding■the■resulting■factorised■expression■using■the■FOIL■method.
factorising by grouping terms■■ If■an■algebraic■expression■has■4■terms■and■no■common■factor■in■all■the■terms,■it■may■be■possible■to■group■the■terms■in■pairs■and■find■a■common■factor■in■each■pair.
Factorise each of the following expressions by grouping the terms in pairs.a 5a + 10b + ac + 2bc b x - 3y + ax - 3ay c 5p + 6q + 15pq + 2
think Write
a 1 Write■the■expression. a 5a■+■10b■+■ac■+■2bc
2 Look■for■a■common■factor■of■all■4■terms.■(There■isn’t■one.)■If■necessary,■rewrite■the■expression■so■that■the■terms■with■common■factors■are■next■to■each■other.
3 Take■out■a■common■factor■from■each■group. =■5(a■+■2b)■+■c(a■+■2b)
4 Factorise■by■taking■out■a■binomial■common■factor. =■(a■+■2b)(5■+■c)
WorkeD exAmple 21
WorkeD exAmple 22
number AnD AlgebrA • pAtterns AnD AlgebrA
86 maths Quest 9 for the Australian Curriculum
b 1 Write■the■expression. b x■-■3y■+■ax■-■3ay
2 Look■for■a■common■factor■of■all■4■terms.■(There■isn’t■one.)■If■necessary,■rewrite■the■expression■so■that■the■terms■with■common■factors■are■next■to■each■other.
3 Take■out■a■common■factor■from■each■pair■of■terms. =■1(x■-■3y)■+■a(x■-■3y)
4 Factorise■by■taking■out■a■binomial■common■factor. =■(x■-■3y)(1■+■a)
c 1 Write■the■expression. c 5p■+■6q■+■15pq■+■2
2 Look■for■a■common■factor■of■all■4■terms.■(There■isn’t■one.)■If■necessary,■rewrite■the■expression■so■that■the■terms■with■common■factors■are■next■to■each■other.
=■5p■+■15pq■+■6q■+■2
3 Take■out■a■common■factor■from■each■pair■of■terms. =■5p(1■+■3q)■+■2(3q■+■1)=■5p(1■+■3q)■+■2(1■+■3q)
4 Factorise■by■taking■out■a■binomial■common■factor. =■(1■+■3q)(5p■+■2)
■■ The■answers■found■in■Worked■example■22■can■each■be■checked■by■expanding■the■brackets■using■the■FOIL■method.
■■ There■are■only■3■possible■pair■groupings■to■consider■with■this■technique:1st■and■2nd■terms■+■3rd■and■4th■terms■or■1st■and■4th■terms■+■2nd■and■3rd■terms■or■1st■and■3rd■terms■+■2nd■and■4th■terms.
remember
1.■ When■factorising■any■number■of■terms,■look■for■the■highest■common■factor■of■all■the■terms.
2.■ A■binomial■factor■is■an■expression■that■has■2■terms.3.■ The■HCF■of■an■algebraic■expression■may■be■a■binomial■factor,■which■is■in■brackets.4.■ When■factorising■expressions■with■4■terms■that■have■no■highest■common■factor:
(a)■ group■the■terms■in■pairs■with■a■common■factor(b)■factorise■each■pair(c)■ factorise■the■expression■by■taking■out■a■binomial■common■factor.
more factorising using the highest common factorfluenCY
1 We21 Factorise■each■of■the■following■expressions.a 2(a■+■b)■+■3c(a■+■b) b 4(m■+■n)■+■p(m■+■n)c 7x(2m■+■1)■-■y(2m■+■1) d 4a(3b■+■2)■-■b(3b■+■2)e z(x■+■2y)■-■3(x■+■2y) f 12p(6■-■q)■-■5(6■-■q)g 3p2(x■-■y)■+■2q(x■-■y) h 4a2(b■-■3)■+■3b(b■-■3)i p2(q■+■2p)■-■5(q■+■2p) j 6(5m■+■1)■+■n2(5m■+■1)
2 We22 Factorise■each■of■the■following■expressions■by■grouping■the■terms■in■pairs.a xy■+■2x■+■2y■+■4 b ab■+■3a■+■3b■+■9c xy■-■4y■+■3x■-■12 d 2xy■+■x■+■6y■+■3
exerCise
3h
eBookpluseBookplus
Activity 3-H-1Reviewing HCF in
groupsdoc-3999
Activity 3-H-2Using HCF in groups
doc-4000
inDiViDuAl pAthWAYs
(a■+■b)(2■+■3c) (m■+■n)(4■+■p)(2m■+■1)(7x■-■y) (3b■+■2)(4a■-■b)
(x■+■2y)(z■-■3) (6■-■q)(12p■-■5)(x■-■y)(3p2■+■2q) (b■-■3)(4a2■+■3b)(q■+■2p)(p2■-■5) (5m■+■1)(6■+■n2)
(y■+■2)(x■+■2) (b■+■3)(a■+■3)(x■-■4)(y■+■3) (2y■+■1)(x■+■3)