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


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


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

14

7145

1

3

29

9

1623

x + 2

2x■-■5 2



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

WorkeD exAmple 1


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

WorkeD exAmple 2



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

WorkeD exAmple 3


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)


=■200■voltsThe■voltage■is■200■volts.

b 1 Write■the■formula. b V■=■IR

2 Substitute■V■=■300■and■R■=■600. 300■=■I(600)


I■=■300600

■=■12■ampere

The■current■is■12■ampere.

WorkeD exAmple 4



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


=■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


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

WorkeD exAmple 5

exerCise

3A

3 7 -2 -8 1

-113

1

2- 1

4-1

9


57Chapter 3 Algebra

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

doc-3978

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



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


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.)

eBookpluseBookplus

Digital docWorkSHEET 3.1

doc-6138

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



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.


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.)

exerCise

3b

eBookpluseBookplus

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

eBookpluseBookplus


doc-6123

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



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


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



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


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

exerCise

3C

eBookpluseBookplus

Activity 3-C-1Reviewing algebraic

operationsdoc-3984

Activity 3-C-2Simplifying algebraic

operationsdoc-3985

Activity 3-C-3Applying algebraic

operationsdoc-3986

inDiViDuAl pAthWAYs

9ab,■-ab

4x

-3za,■-az

2x2,■-x2

-2x2y-x2y5,■-3x2y5

p3x2w5,■-5x2p3w5

4y5z4x2,■-2x2z4y5

7x 11y 19m11q 8r 3x

8a 8y 13x



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


doc-6126

eBookpluseBookplus


doc-6125

eBookpluseBookplus


doc-6140

eBookpluseBookplus


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eBookpluseBookplus


doc-6128

eBookpluseBookplus

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


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



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


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



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


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

eBookpluseBookplus

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



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)

eBookpluseBookplus


doc-6125


eBookpluseBookplus


doc-6127


eBookpluseBookplus


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■+■

2b2 ■

+■2b

c

k

3p■-

■9pr

■-■2

q■+■

6qr

l7c

d■-■

35c■

-■2d

2 ■+■1

0d

m

12x

2 ■-■

7xy■

+■y2

n2p

2 ■-■

rp■-

■2pq

■+■q

r

o

15k■

+■5■

-■6j

k■-■

2j

9

a■xz

■+■x

■+■y

z■+■

y

b

pr■+

■3p■

+■qr

■+■3

q

c

2xz■

+■8x

■+■y

z■+■

4y

d3p

r■+■

3p■+

■qr■

+■q

ea2 ■

+■3a

b■+■

2b2

f

2c2 ■

-■5c

d■-■

3d 2

g2x

2 ■-■

xy■-

■3y2

h

4p2 ■

+■pq

■-■3

q2■


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



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.


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



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


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


4y

y2■-■x2■=■(y■+■x)(y■-■x)■ Difference■of■two■squares.



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


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


doc-6143


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



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


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



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


=■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


=■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)


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


doc-6130

eBookpluseBookplus


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

4,■-

7,■-

2,■-

1,■1

,■2,■7

,■14

e-8

,■-4,

■-2,

■-1,

■1,■2

,■4,■8

f-1

00,■-

50,■-

25,■-

20,■-

10,■-

5,■-

4,■-

2,■-

1,■1

,■2,■4

,■5,■1

0,■2

0,■2

5,■

50,■1

00g

-4

2,■-

21,■-

14,■-

7,■-

6,■-

3,■-

2,■-

1,■1

,■2,■3

,■6,■7

,■14,

■21,

■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)



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


-3(ab■-■6m■+■7)


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)


=■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)


=■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



b 1 Write■the■expression. b x■-■3y■+■ax■-■3ay


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


=■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)

3a using pronumerals 3b 3c 3d 3e contents 3f 3g 3h 3i · 52 maths quest 9 for the australian...

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