Algebra Chapter 02

7/26/2019 Algebra Chapter 02

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Patterns and algebra

Algebra2

Algebra has helped people unlock some of the secrets of the universe.

Algebraic skills are fundamental to solving equations. Many natural

phenomena, such as rainbows (caused by the diffraction of light in

raindrops), reflections in water or mirrors, and sound waves, can be

explained and described using algebra and equations.


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understand and use variables, algebraic terms and expressions

recognise and combine like terms to simplify algebraic expressions

simplify algebraic expressions that involve multiplication and division substitute into algebraic expressions

expand and factorise algebraic expressions

simplify algebraic expressions involving fractions.

algebraThe branch of mathematics that uses symbols to describe thenumber patterns and relationships occurring in the real world.

evaluate Find the value of an algebraic expression or formula.

simplifyTo make shorter or less complex. When the fraction issimplified, the result is . In algebra, 4a+ 3asimplfies to 7a.

substitution Replacing a variable with a number in an algebraicexpression.

expand Rewrite an expression such as 3(2k+ 7) without groupingsymbols. When 3(2k+ 7) is expanded, the result is 6k+ 21.

factorise To rewrite an expression with grouping symbols, by taking outthe highest common factor. Factorising is the opposite of expanding.

The speed (in km/h) of a car after tseconds is given by the expression

100 6t. Is the speed of the car increasing or decreasing? Can you

calculate the speed of the car after 7 seconds? What is the cars speed after

10 seconds?

In this chapter you will:

Wordbank

3

12------

1

4---

Think!

A LGEBR A 41 CH AP TER 2


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42 N EW CEN TUR Y M ATH S 9 : S TAGES 5 .2/5 .3

Generalised arithmeticIn arithmeticwe use symbols to represent numbers when we know their value. These symbols

are called numerals.

The symbol 6 is used to represent the number of dots in this circle.

The symbol 4 is used to represent the number of crosses in this rectangle.

In algebrawe use symbols to represent numbers when we do notknow their value or when

their value may change. Letter symbols are commonly used (a, b,x,y) but any symbol may be

used. These symbols are called variables(or sometimes pronumerals).

Algebra is used to write generalisations. For instance, the expressionx+8 is a general way ofstating any number plus 8, where the variablexstands for any number.

1 Simplify where possible:

a 9yy b 9y9 c 8ab+2ab

d 8a+5y+a e 7ab+2ab f 16 +xy+10

g 3y2+2y h 18ay+2ay i x+5x+5

j 20abc2

3ab2c k 9m

2

5m3m2

l 2p2q+4pq

2

7p2q

2 Simplify:

a 9 2xy b 3 2w2 c 6a2p

d 5m10m e 8y2y f 3a5ac

3 Simplify:

a b 8xy2x c 20abc5c

d e 12a2(2a) f

4 Ifp=6, find the value of:

a p2 b 6p c 9 p

d 3p+20 e f 12 +2p

5 Write the factors of:

a 12 b 27 c 9x d 16ab

6 Find the highest common factor (HCF) of the following:

a 18 and 10 b 50 and 20 c 24 and 27

d 8xand 5x e 16aband 8a f 20 and 4y

g 30mpand 12m h 18xyand 4w i 20ab2and 8c

7 Evaluate:a + b c d

e f g + h

16a

4---------

3ay

6---------

40ab c2

8ac------------------

3p

4------

3

5---

2

3---

9

10------

1

3---

5

8---

10

13------

2

3---

3

5---

5

8---

1

4---

2

5---

7

8---

3

10------

5

6---

5

3---

9

20------

Start up

Worksheet2-01

rainstarters 2

Skillsheet2-01

Algebraicexpressions

Worksheet2-03

Generalisedarithmetic
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A L GEB RA 43 CH AP TER 2

The average,x, of the two numbers 5 and 9 isx= . So, in general, we can say that the

average,x, of any two numbers, aand b, isx= .

These examples show that algebra can be used to make generalisations. The generalisations

are algebraic expressions that show the steps we need to take in order to obtain a certain result.

Simplifying algebraic expressionsAn algebraic expressionis an expression or statement consisting of algebraic terms. Forexample, 3d+7k5 and 15hk25 +xare algebraic expressions.

Algebraic expressions should always be simplified. This means that the algebraic expression

should be written in as short a form as possible.

Adding and subtracting algebraic expressionsLike termsare terms in which the variables are exactly the same.

For instance:

3kand 7kare like terms 2dand 5mare notlike terms because they do not have exactly the same variables

3aband 7baare like terms, because abis the same as ba

2xyand 5yzare not like terms

2xand 3x2are not like terms.

5 9+

2------------

a b+

2------------

Add and subtract like termsonly.

Patterns and pineapples

The pattern of Fibonacci numbers is 1, 1, 2, 3, 5, 8, 13, 21,

Surprisingly, perhaps, these numbers often appear in nature.

For example, a pineapple is covered by scales, each of which

is a hexagon. Any one of these hexagons is on three different

spirals on the pineapple. The number of hexagons on each

of these spirals is 8, 13 and 21 (the steepest spiral).

These numbers are three consecutive Fibonacci numbers!

1 Find three other examples of Fibonacci numbers appearing in nature.2 Can the Fibonacci number pattern be represented algebraically?

the three different spirals

Just for the record Worksheet2-02

Investigatingpatterns in

numberproblems
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1 Simplify, where possible:

a 8y+5y b 12x2x c 13 +4y d 7a+2aa

e 8e8e f 19yy g 9y3+2y2 h 13n+2nn

i 5y6y j 6ac +2ca k 8xyxy2yx l 8e2f+2fe2

2 Simplify, where possible:

a 8y+2y+6 b 3a+10 a c 7m+5y2m

d 16 +5x+x e 9ac+2a+ac f 7xy+5 +xy

g 1 +2y+6 h 3ay+8y+ay i x6 5x

3 Simplify each of the following expressions:

a 4m+3 +2m+7 b 2y+6gy3g c 7p+8h6ph

d 8 +3k+7 2k e 2p3 12p+5 f 12y5k+3y4k

g 8t3y2t4y h 6 4n+5 3n i 5a+3f7a4f

j 3 +4w+8 +7w k 3e7t+5e+2t l 4ab+3q5ba7q

m 4c2+3c+5c29c n 5x2+2x33x27x3 o mab3m+2ab

p 4hy23h2hy24h q y10xx+2y r 4a26y10y5a2

4 Simplify each of the following expressions by collecting like terms.

a 5a3a b 7m2+2m c 5k11k

d 3uu e 2d27d2+3d f 5m2m+3t

g 8k+2y+5k h 7 3r+6 i 9b3u5b

j 8e+5 3e+7 k w+7w l 4t+3g+5t7g

m 9y27 2y213 n 5mn+3mn o 3 +5f +7 6f

p 3 5f+7 6f q 8ab2b4ba+5a r 3 8l4 4l

s 5h+6 +5h t n3n+2 u 6p3+3p+5p+2p3

v s2+ss2+s w 3 2cd+5cd1 x 8k3mm2+2k

5 Find the perimeter (distance around the outside) of each of the following figures:

a cb

40h

9h

41h

2a

5c

9c

4a

4c

6a

2p

p+2

p+2

2p1

3x

2p1

p+4

x+4

3p+5

d e

2xy

y+x+1

f

Exercise 2-01

Skillsheet2-02

Algebra usingdiagrams

CAS2-01

Collecting liketerms

SkillBuilder8-058-06Introductory

algebra
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Multiplying and dividing algebraicexpressionsWhen multiplying algebraic expressions, we use shorthand notation. So, instead of writing

4 ab, we write 4ab.

When dividing, the algebraic expression is usually written as a fraction. So the expression h2

is written as .

6 The dimensions of this diagram are given in terms of

mand n.

Write, in terms of mand n, expressions for:

a the length of CD

b the length ofBC

c the perimeter of the figure.

7 The dimensions of this diagram are given in terms of

uand t.

Write, in terms of uand t, expressions for:

a the length of CD

b the length ofBC

c the perimeter of the figure.

8 The perimeter of each shape is (4a+2c) units. Find the length of the sideAB.

3m

3n

m5n

B C

A F

E D

A B

C D

EF

3u+2t

2u+5t

ut

u+t

A

C

B

2a+c

a+c

a b c

A

CB

D

a+c+2

a+1

A B

CWorksheet

2-04Perimeter and

area

h

2---

Example 1

Simplify:

a 5 10y b 2y4w c 5abac

Solutiona 5 10y=50y b 2y4w=8yw c 5abac=5a2bc

Simplify:

a a2b4ab b 27cf(3fc) c 5p215p d 18xy210yx2

Example 2


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Solution

a a2b4ab= b 27cf(3fc) =

= =

=

9

c 5p215p= d 18xy210yx2 =

= =

Simplify:

4x+9x23x.

SolutionUse the rule for the order of operations when simplifying.

a 4x+9x23x=4x+

=4x+3x

=7x

a a b

4 a b---------------------

1

1 1

1

3 fc27 c f------------------------

1

1 11

1-9

a

4---

1------9

5 p p

15 p----------------------

13

1 118 x y y

10 y x x---------------------------------

1

1 15

19

p

3---

9y

5x------

Example 3

9x x

3 x---------------

1

1 1

3

1 Simplify each of the following:a 3 5p b 9x2 c 4 2m3

d 2h4y e 3y2q f 5y2y

g 3w4 h 4n2d i 6p(4p)

j 2a3 4b k 3x4y(5k) l 5r(2r) (3)

m 4c2f3c n 6t3t2 o 4hk3k

p 2ce4ec q 3a2d2d r 5m2n3mn

2 Simplify each of the following:

a 12y4 b 15kk c 24a6a d 9mk3m

e 15uy5u f 36mnp6mp g 16a(4) h 25ab(5b)

i abcbc j 18xy(9y) k 5m2m l 28c2f4cfm 40w210w n a2b23ab2 o 9yx218xy p 4ab12a

q 3x26xy r 4ab2c20abc s 9m312m4 t 25ab215a2b3

u a2bc5b2ac2 v 14xyw(6y2) w 8a2c12abc2 x 8mp29m2p

Exercise 2-02

Example 1

Example 2
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Substitution in algebraic expressionsIn any algebraic expression, we can replace the variables with numbers to obtain a value for

the expression. This process is called substitution.

For instance, the perimeter, Pmetres, of a rectangle of length land

breadth bmetres can be expressed algebraically as:

P=2l+2b

When l=4.5 and b=2.8, Pcan then be found as follows:

P =2 4.5 +2 2.8

=14.6

So the perimeter is 14.6 metres.

3 Use the order of operations to simplify the following:

a 5 4a(10a8a) b 5 4a2a c 6xy3 5xy

d 2y5 2y e yy4 f m2m5

g 9m2+5m2m h d4 5d i (11ac+ac) (6a2c)

j 2m26m4m k (r 3r) (10r25r) l 9y8a2y4a

m 6c+15c3 n 3x2x5x o 2gg15g35g

4 Find the area of each of the following shapes:

5 Find an expression for the volume of each of the following prisms:

2d

3a

a b c

4p

6y c

3c4k

2k

2m

2m

2m

4y

5d

3t

6k

2w

w

a b c

Example 3

Worksheet2-04

Perimeter andarea

Worksheet2-05

Bits and pieces

CAS2-02

Multiplying anddividing

expressions

Worksheet2-06

Tables of values

lm

bm
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Substitution

A spreadsheet can be used to evaluate an algebraic expression for different values of variables.

Step 1: To find the value of P=x2+3x2 for ten values ofx, set up your spreadsheet as

follows:

Step 2: Using Fill Down, copy the formulas down to A12 and B12.

Step 3: Print out your spreadsheet and paste it in your workbook.

A B

1 x P

2 5 =A2^2+A2*32

3 =A2+1

Using technology

Example 4

If t=4, c=2 and h=5, find the value of each of the following:

a 2hc+t b h2+c2 c 3t2

Solutiona 2hc+t b h2+c2 c 3t2

=2 5 (2) +4 =52+(2)2 =3 42

=16 =29 =3 16

=48

IfE=mgh,findEwhen m=10, g=9.8 and h=23.4.

SolutionE=mgh

=10 9.8 23.4

=2293.2

Example 5

1 Find the value of 2p+qwhen:

a p=1, q=2 b p= , q=1 c p=5, q=8

2 Find the value of when:

a x=40,y=5, w=15 b x=8,y=10, w=2

c x=6,y=12, w=3 d x=15,y=9, w=10

3 Find the value of abc2when:

a a=12, b=3, c= b a=5, b=2, c=1

c a=3, b=10, c=0 d a= , b= , c=4

1

2---

x y

w-----------

1

6---

3

4

--- 1

2

---

Exercise 2-03Example 4

CAS2-03

ubstituting inton expression
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Using grouping symbols in algebraGrouping symbols may be used in algebra in a similar manner to the way they are used in

arithmetic.

For example, 3(4 +7), which can also be written as 3 (4 +7), can be evaluated in two ways.

3(4 +7)=3 (4 +7) or 3(4 +7)=3 (4 +7)

=3 11 =3 4 +3 7

=33 =12 +21

=33

4 Find the value ofpqqtwhen:

a p=3, q=2, t=6 b p=4, q=2, t=1

c p=8, q= , t=10 d p= , q= , t=

5 Evaluate:

a + when a=20, c=10, d=5

b xyw2whenx=2,y= , w=2

c wheny=8, a=12,x=6, b=2

6 Evaluate each of the following if m=2, n=3,p=5:

a m+n+p b mn c np d m2n2

e npm f mn+p g 5n+3p h

i j 5m2 k m2+p4n l

7 Ifp=2, q=5 and r=8, evaluate each of:

a p+qr b 4pr c 3(p+r+q) d 2p3q

e pr2 f pqr g q22q h (p+q)2r2

i (3 r)2 j + k l

8 a If P=2L+2B, find PwhenL=4.7 andB=2.5.

b If V=LBH, find Vgiven thatL=4,B=6.5 andH=3.5.

c IfA= bh, findAif b=15 and h=12.

d IfA= h(a+b), findAgiven that h=4.5, a=6 and b=8.

e Given thatA= xy, findAwhenx=4.8 andy=8.4.

1

2---

1

3---

2

5---

2

3------

a

c

--- c

d

---

1

4------

y a

x b-----------

m n p+ +

p n+

m------------

p

n m-------------

1

2---

p

q---

r

q---

qr

p-----

p r

q-----------

1

2---

1

2---

1

2---

Example 5

Worksheet2-07

Substitutionpuzzle

Working mathematically

Communicating and reasoning: Grouping symbols and areas

Consider the rectangle below, which has length (r+t) units and breadth 3 units.

1 Write an expression for the area of the rectangle.

2 The rectangle has been divided into two smaller

rectangles, A and B. The lengths of A and B are runits

and tunits respectively, and the breadth of both is

3 units. Write the area of A and B in simplest form.3 Write the sum of the areas of A and B.

r + t

3 A B


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Expanding algebraic expressionsWhen expanding an expression, each term inside the grouping symbols (brackets) is multipliedby the term outside the grouping symbols.

4 Since the area of the whole rectangle is the same as the sum of the areas of A and B, we

can write 3(r+t) =3r+3t. Notice that each term inside the grouping symbols is

multiplied by the term outside the grouping symbols.

Repeat this process for rectangles with the following measurements:

a lengthx+yand breadth 7 b length 3 +kand breadth 4

c length a+band breadth c

(Compare your results with those of other students.)

5 Use the rectangle shown to help you simplify the area of A, 4(3 k).

3 k k

4 A

3

B

Example 6

Expand each of the following:

a 5(3 d) b a(3 +2a) c 5f(2f+3)

Solution

a 5(3 d) b a(3 +2a) c 5f(2f+3)

= 5 3 5 d =a3 +a2a =5f2f+5f3= 15 5d =3a+2a2 =10f2+15f

Expand each of the following:

a 3g(2 +5g) b 4(3 5x)

Solutiona 3g(2 +5g) b 4(3 5x)

= 3g2 +(3g) 5g = 4 3 (4) 5x

= 6g+(15g2) = 12 (20x)

= 6g15g2 = 12 +20x

Expand each of the following expressions:

a (7 +3w) b (h2m)

Solutiona (7 +3w) b (h2m)

=1(7 +3w) =1(h2m)

=1 7 +(1) 3w = 1 h(1) 2m

=7 +(3w) = h(2m)

= 7 3w = h+2m

Example 7

Example 8


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Example 9

Expand and simplify:

a 5(1 +2k) 7k b 4(x+3) +5(2x7)

c 6(1 2c) 2(3c+1) d 3p(p4) 2p(4p5)

Solution

a 5(1 +2k) 7k b 4(x+3) +5(2x7)=5 1 +5 2k7k =4 x+4 3 +5 2x5 7

= 5 +10k7k =4x+12 +10x35

= 5 +3k =14x23

c 6(1 2c) 2(3c+1) d 3p(p4) 2p(4p5)

= 6 1 6 2c2 3c+(2) 1 =3pp3p4 2p4p2p(5)

= 6 12c6c2 =3p212p8p2+10p

= 4 18c(or 18c+4) =5p22p

1 Expand each of the following:

a 5(t3) b 4(1 3b) c 4(3k+5)

d 2(d+e) e m(n+p) f 5(1 2w)

g c(2c+d) h 4(8 7f) i 3k(k2w)

j a(a3) k 7(2p3m) l 4n(3n+5y)


a 3(s+2) b 5(4f+3) c 2(2m+c)

d 4(3d+5e) e w(2w5) f 7(5 a)

g 7(3b2q) h 8(2d+3) i 4r(r5y)

j 3(3 5m) k p(2p+p2) l 2v(3 7n+3r)

3 Expand each of the following expressions:

a (4 +3m) b (5x+2) c (3f+5)

d (8 +y) e (1 +p) f (6n1)

g (1 2a) h (1 e) i (10 5q)

4 Expand and simplify:

a 4k+3(2k+5) b 3(1 5y) +6y c 4(m+3n) +2n

d 6 3(4 +x) e 8 5(2d7) f 12p4(1 3p)

g 3w2(4 +2w) h 5(t+4) +3(t+2) i 2(3v+1) +5(v+7)

j 4(3

2h)+

2(3

4h) k 4(2m+

5)

2(m+

4) l 5(6+

d)

4(3d+

1)m 3(4 +3e) 4(2e1) n 8(3c+1) 5(2c3) o 2(1 8w) 3(1 2w)


a 5m(1 +2m) +3m(m+1) b 2w(3w+2) w(3w+4) c 4r(2r+1) 5r(r+2)

d 3q(q+4) q(q1) e 5v(2v+3) 2v(3v4) f 2a(4a1) 3a(2a5)

g 5t(4 3t) 2t(1 6t) h m(m3) m(2m+4) i 4e(1 2e) e(3e+4)

j 5(6d3) (4d+3) k 2(1 +2h) (1 5h) l 7r(1 2r) (3r4)

m 2k(k1) (4k+5) n 3f(5f+2) 4f(2f1) o 3b(1 4b) +2(8b4)


a 2w+(3w+5) b 3(2k5) +7 c 2(3r+2) 9r

d 7 2(3m+2) e 2(h+3) +3(h4) f 5x3x(2 x)


Example 7

Example 8

Example 9

CAS2-04

Expandingexpressions


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FactorsA factoris a number or algebraic expression that divides evenly intoanother whole number

or algebraic expression.

For example, 5 is a factor of 30 (because 30 5 =6), and 4pis a factor of 20pw(because

20pw4p=5w).

The relationship between products and factors is shown below.

5 2 =10, so we say that 5 and 2 are factors of 10.

1 10 =10, so 1 and 10 are also factors of 10.

The factors of 10 are 1, 2, 5, 10.

5aay=5a2y, so 5aand ayare factors of 5a2y.

5ya2=5a2y, so 5yand a2are also factors of 5a2y.

5a2y=5a2y, so 5a2andyare also factors of 5a2y.

The factors of 5a2yare 1, 5a2y, 5, a2y, a, 5ay, 5a, ay, 5a2,y, 5y, a2.

g 1 4(s1) h 3 (4 7e) i 5q2(3 2q) +5

j 4(e+3) 3(3e4) k 5(2 b) (b+1) l 4(3de) 2(d+3e)

m 3(4 2y) +2(5 2y) n 8g3(1 2g) +5 o t(t+h) +h(t+h)

7 a The profitD, in dollars, is given by the expression 5(3d+1) 2(2d1). Simplify this

expression.

b The profit P, in dollars, for selling ice creams is 1.55(4n1) +0.85(n+1), where nis thenumber of ice creams sold.

i Simplify this expression.

ii If 200 ice creams are sold, what is the profit?

c The profit per TV set (in dollars) is given by 25(N+250) +45(N150), whereNis the

number of TV sets sold.

i Simplify this expression.

ii IfN=10, what is the profit?

Multiplying by 9, 99, 199,

When multiplying by 9, 99, 199, remember that:

9 =10 1, 99 =100 1, 199 =200 1,

1 Examine the following examples:

a 37 9 b 99 85 c 12 199

= 37 (10 1) = 85 (100 1) = 12 (200 1)

= 37 10 37 1 = 85 100 85 1 = 12 200 12 1

= 370 37 = 8500 85 = 2400 12

=

333 =

8415 =

23882 Find answers for the following:

a 87 9 b 45 99 c 9 76

d 89 99 e 8 199 f 14 199

g 75 999 h 28 99 i 9 895

j 99 95 k 55 99 l 199 18

Skillbank 2

SkillTest2-01

Multiplying by9, 99 or 199

SkillBuilder7-057-17

Factors
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Factorising algebraic expressionsWhen the expression 4(2y+5) is expanded, the result is 8y+20. Factorisingis the reverse of

expanding.

Factorisation breaks an expression into factors. When 8y+20 is factorised, the result is 4(2y+5).

The factors are 4 and (2y+5).

Common factorsA common factoris a number that is a factor of two or more numbers or algebraic expressions.

For example, 3kis a factor of both 15k2and 30kh(because 15k23k=5k, and 30kh3k=10h).

The first step in factorising an algebraic expression is to determine the highest common factor

(HCF)of the terms in the expression.

Expand

(remove brackets)

4(2y+5) 8y+20

Factorise

(insert brackets)

Example 10

Factorise each of the following:

a 5t+10 b 24x216xy

Solutiona The HCF of 5tand 10 is 5.

5t+10=5 t+5 2 (rewriting the expression using the HCF)

=5(t+2)b The HCF of 24x2and 16xyis 8x.

24x216xy=8x3x8x2y

=8x(3x2y)

Factorise each of the following:

a 3m+9 b 5y225y

Solutiona 3 is the highest common factor of 3mand 9.

3m+9 =3 m+(3) (3)=3[m+(3)]

=3(m3)

b 5yis the highest common factor of 5y2and 25y.

5y225y=5yy(5y) (5)

=5y[y(5)]

=5y(y+5)

Example 11

Skillsheet2-03HCF by

factor trees

1 Factorise each of these fully:

a 9m+12 b 3t+6 c 4w12 d 6k24

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Adding and subtracting algebraic fractionsWhen adding or subtracting any fractions, the denominators must be the same. (Never add or

subtract the denominators.)

e 20c+15 f 9k27 g 16 24w h 12e+18

i 15 +20p j 10 +12t k 24y30 l 2q22

m 18g+27 n 16w8 o 9v+33 p 28 21t

2 Factorise the following expressions:

a hkh b wy+y c mn3m d 6p+pr

e ab+ac f 4a+12b g x2

+x h 5 5qi 12y+18x j 2y+y2 k d2+8d l 3e2+6e

m 12m210m n a2+ac o 5n220n p 7g+9g2

3 Factorise each of these completely:

a 16x212xy b 8p2r+6pr c 4m2h24m

d 14hk+21ky e 28wv21v2 f 18abc+24ab

g 4mn26mn h 8rt18r2 i 9a2y+15y2

j 20wy+18wy2 k 15 30b2 l 8wxh+4x2h

m 9e+12 +6e2 n 15hk3h2+9h o 9ef12e+6f2

4 Factorise each of the following:

a 4p8 b 9k+27 c 3g+6 d 18w+12e 6x+4 f 12m16 g 20e15 h 18n21

i x27x j 8w224h k ap+aq l 24xy16x

5 Factorise each of these fully:

a d2+dh b 2p220p c 4y16 d trt

e 16k2t2+4kt f 10ab+20a g 9 3w h p+ q

i x2 xy j k kt k 9xy2+12xy l 24m2+6m

6 Factorise each of these completely:

a 8a+12y+4 b 20xy10x2+5y c t2m+tm2+tm

d 5(a+6) +h(a+6) e 7y(m+2) p(m+2) f c(a+b) d(a+b)g 10p210pq h w(d+3) +4(d+3) i 5a(xy) +10ay(xy)

1

2---

1

2---

1

4---

1

4---

3

4---

1

4---

CAS2-05

Factorisingexpressions

Skillsheet2-04

ctorising usingdiagrams

Example 11

Example 12

Simplify each of the following:

a + b + c d

Solution

a + b +

= =

c d

= =

=

r

3---

r

3---

3y

5------

4y

5------

p

8---

5

8---

7m

10-------

3m

10-------

r

3---

r

3---

3y

5------

4y

5------

2r

3-----

7y

5------

p

8---

5

8---

7m

10-------

3m

10-------

p 5

8------------

4m

10-------

2m

5-------
http://../technology/texas/cas_0205.tiihttp://../technology/texas/cas_0205.tiihttp://../skillsheet/skillsheet_0204.pdfhttp://../skillsheet/skillsheet_0204.pdfhttp://../technology/texas/cas_0205.tiihttp://../skillsheet/skillsheet_0204.pdf


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If the denominators are not the same when adding or subtracting fractions, then we must convert

them so that the fractions have a common (the same) denominator. A quick way of finding a

common denominator is to multiply the denominators of both fractions together.

Example 13


a + b + c d +

Solution

a + b + c d +

= + = + = = +

= + = + = = +

= = = =

Simplify .

Solution

=

=

=


a 5a4c+ c b

Solution

a 5a4c+ c

= 5a 4c+ c

= +

= +

= +

=

b

=

=

=

=

=

k3--- k

2--- 3h

4------ 2h

5------ 5m

8------- 2m

3------- 3

4--- 2a

3------

k

3---

k

2---

3h

4------

2h

5------

5m

8-------

2m

3-------

3

4---

2a

3------

2 k

2 3------------

3 k

3 2------------

5 3h

5 4---------------

4 2h

4 5---------------

3 5m

3 8-----------------

8 2m

8 3-----------------

3 3

3 4------------

4 2a

4 3---------------

2k

6------

3k

6------

15h

20---------

8h

20------

15m

24----------

16m

24----------

9

12------

8a

12------

5k

6------

23h

20---------

m

24-------

9 8a+

12---------------

Example 14

4

c---

7

3c------

4

c---

7

3c------

3 4

3 c------------

7

3c------

12

3c

------ 7

3c

------

5

3c------

Example 15

3

4---

7a

2------

m 3+

5-------------

m 2

4-------------

3

4

--- 7a

2

------

7a

2------

3

4---

10a

2---------

7a

2------

16c

4---------

3c

4------

10a 7a

2---------------------

3c 16c

4---------------------

3a

2------

13c

4-----------

3a

2------

13c

4---------

m 3+

5

------------- m 2

4

-------------

4 m 3+( )

4 5---------------------------

5 m 2( )

5 4---------------------------

4 m 3+( )

20----------------------

5 m 2( )

20---------------------

4 m 3+( ) 5 m 2( )

20-------------------------------------------------

4m 12 5m 10++

20----------------------------------------------

22 m

20----------------


17/22

56

N EW CEN TUR Y M ATH S 9 : S TAGES 5 .2/5 .3

Multiplying and dividing algebraicfractions Remember:

when you are multiplying and dividing fractions, the denominators do not need to be the same

when multiplying fractions, cancel any common factors, then multiply numerators and

denominators when dividing by a fraction , multiply by its reciprocal .


a + b + c d

e + f g h +

i + j k l +

m + n o p


a + b c d

e + f + g + h

i j k + l +

m + n + o p +


a + b c d

e + f g + h +

i j k + l

4 Simplify:

a 5k +2m b 3y2m + c +

d + e 2k+5h f +

g h +8e3t i

j + k l

m n o

p q + + r +

s + + t u + 2y

w

4----

2w

4-------

5k

8------

3k

8------

7m

10-------

2m

10-------

x

3---

2x

3------

4

q---

5

q---

5

3d------

2

3d------

4r

w-----

r

w----

2

c---

5

c---

4t

3-----

s

3---

10y

3h---------

7y

3h------

6

a---

5

a---

4

e 1+------------

6

e 1+------------

p

z---

3

z---

5u

8g------

3u

8g------

4

9f------

1

9f------

7e

8------

3e

8------

x

3---

x

4---

s

3---

s

7---

h

5---

h

3---

m

7----

m

2----

w

4----

w

5----

5t

4-----

2t

5-----

2p

5------

p

3---

5r

2-----

5r

3-----

3c

2------

c

5---

2d

4------

r

5---

3h

5------

2a

3------

5

6---

4w

5-------

34--- 2a

7------ a

2--- b

3--- 7e

8------ 2e

3------ 3

5--- 4k

3------

2

3p------

5

p---

4

k---

3

2k------

5

4m-------

2

m----

10

3y------

7

y---

4

5m-------

1

m----

8

x---

5

3x------

4

a---

2

5a------

k

4c------

2

c---

a

4m-------

b

m----

7

8y------

3

2y------

8

3d------

3

5d------

2a

w------

c

4w-------

k3--- m

4---- 2y

5------ m

3---- x

5--- x 2+

3------------

y 3+

4------------

y 1

3-----------

3h

4------

4k

5------

x 1

4-----------

x 2

5-----------

m 6+

5-------------

m 2+

3-------------

5t

4-----

7e

2------

k 3+

10------------

k 1

7-----------

m 1

4-------------

m 2

3-------------

2k 1+

3---------------

k 1

2-----------

3x 1

4---------------

2x 9

5---------------

1 m

5-------------

m 2

7-------------

9 2k

3---------------

k 5

2-----------

2m 5+

4-----------------

m 1

9-------------

8 3x

2---------------

2x 5+

3---------------

x

4---

x 2+

3------------

2x

5------

w 1+

7-------------

3w

2-------

w 1

5-------------

x 12

----------- x 23

----------- x 34

----------- r 45

----------- r 23

----------- r 12

----------- y 7+2

------------ y3---


Example 13

Example 14

Example 15

CAS2-06

Adding andsubtractingalgebraicfractions

a

b---

b

a---


18/22


Example 16


a b c

Solution

a b c

= = =

= = =

= 1

Simplify each of the following:a b

Solution

a b

= =

= =

3

d---

4

c---

4

k---

3k

16------

5

2r-----

8r

11------

3d--- 4

c--- 4

k--- 3k

16------ 5

2r----- 8r

11------

3 4

d c------------

4

k---

3k

16------

1

1

1

4

5

2r----

8r

11----

1

4

12

cd------

3

4---

20

11------

9

11------

Example 17

3

h---

4

k---

xy

5-----

3x

25------

3

h---

4

k---

xy

5-----

3x

25------

3

h---

k

4---

xy

5----

25

3x-----

1

1

5

1

3k

4h------

5y

3------

1 Simplify each of the following products:

a b c d e

f g h i j

k l m n o

2 Simplify each of the following divisions:

a b c d e

f g h i j

k l m n o


a b c d

e f g 5 h 3

w

3----

1

2---

s

5---

t

4---

3

h---

5

6k------

4

m----

3

n---

l

3---

5

f---

1

v---

2

3v------

2

x---

3

x---

a

b---

c

d---

5p

4------

8

15p---------

4ad

9----------

d

16a---------

d

e---

e

g---

4

ak------

3a

5k------

2

3---

9u

10------

u

3---

3

u---

3z

r-----

2r

9dz---------

r

2---

r

5---

m

6----

n

3---

h

2---

h

8---

q

4---

3

4---

3y

5------

2y

d------

4t

9-----

3t

5-----

3

e---

5

e---

3

2a------

b

6a------

5m

n-------

2m

3n-------

h

k---

k

h---

8w

3x-------

2w

9x-------

3s

4-----

6s

11------

t

3---

3t

5u------

3e

7g------

e

14g---------

xh

5------

3h

15------

3p

2------

p2

4-----

2w

7-------

5w

6-------

8y

5------

3

32y2

----------- s

3---

5s

2-----

3s

7-----

xyz----- 5

y--- a

b--- c

b--- b

3a------ 3

5b------ 4t

9-----


Example 17

CAS2-07

Multiplying anddividingalgebraicfractions


19/22


i j 5p k l 14

m n o p

5mn

2d-----------

4d

n------

1

15mn--------------

7

10pt------------

5

2g------

2

g---

7

h---

1

3h------

5ty

k--------

5ky

t---------

6

r---

15

yh------

5r

9-----

6cf

q---------

5

qf------

10c

a---------

5hk2

3------------

15

hk------

6h

k3------

Worksheet2-08

Algebraicfractionspuzzle

1 For each of the following patterns, find:

i the next three terms

ii the rule for the pattern

iii the value of the 20th term.

a 3, 5, 7, 9, b 7, 3, 1, 5, c 5, 8, 13, 20, 29,

2 The cost of several tennis racquets totals $27rq. If one racquet costs $9q, how many tennis

racquets were purchased?

3 Pencils cost mcents each and erasers costs ncents each. If 40 pencils and 50 erasers are

bought, find:

a the total cost in cents b the total cost in dollars.

4 Find the average score of a cricketer whose scores in six innings are 6r, 2r+8, r5, 2r, r5

and 3r.

5 A rectangular garden has its longer sides each 5 m longer than its shorter sides.

a If one side length is kmetres, write two expressions for the other side length.

b Write two different expressions for the gardens perimeter.

6 The area of a rectangle is 20absquare units.

a If the length of the rectangle is 10aunits, find the width.

b Write the lengths and widths of four other rectangles that have

an area of 20absquare units.

7 Evaluate:

a a22b, when a=4 and b=5 b (5t)2, when t=3

c 5t2, when t=3 d x24y2, whenx= andy=1

e , when r=3 and t=7 f f2+2h2, whenf=4 and h=3

g (p+q2)2, whenp=5 and q=3 h , whenp=5 and q=12

8 Simplify:

a + b

c d

e + f +

10a

1

2---

r

2 t----------

p2

q2

+

4

e 1+------------

6

e 1+------------

5

3v------

2

4v------

r 3+

4-----------

2r

5-----

8

3x 3---------------

2

1 x-----------

1

2x

2---------------

5

x 1

----------- 4w

2

3w

1+( )

--------------------- 2w

27w

1+( )

------------------------

Power plus
http://../worksheet/work_0208.pdfhttp://../worksheet/work_0208.pdfhttp://answers.pdf/http://../worksheet/work_0208.pdfhttp://answers.pdf/


20/22


Topic overview Rate your understanding of and ability with the work in this chapter by copying and

completing the following scales. Draw an arrow on each scale to indicate your rating.

a Able to add and subtract like terms, and multiply and divide simple algebraic expressions.

b Understand and use grouping symbols.

c Able to substitute numbers into algebraic expressions or formulas.

d Able to add and subtract algebraic fractions.

e Able to multiply and divide algebraic fractions.

Language of mathsalgebra algebraic expression brackets consecutive

difference evaluate expand expressions

factor factorise like pattern

product pronumeral quotient reciprocal

relationship simplify substitute variable

1 Explain the difference between evaluatingand simplifying.

2 What is a synonym? Write synonyms for the word like. How many of these can be

used in a mathematical sense?

3 Write the meaning of expand when used in the mathematical sense. Write two other

sentences using this word.

4 What does consecutive mean?

5 Which word from the list above means:

a the answer to a multiplication question?

b to replace a variable with a number in an algebraic expression?

6 Write a sentence that uses at least four of the above words.

Worksheet2-09

Algebracrossword

Worksheet2-10

Algebra review

0 1 2 3 4 5

Low High

0 1 2 3 4 5

Low High

0 1 2 3 4 5

Low High

0 1 2 3 4 5

Low High

0 1 2 3 4 5

Low High
http://../worksheet/work_0209.pdfhttp://../worksheet/work_0209.pdfhttp://../worksheet/work_0210.pdfhttp://../worksheet/work_0210.pdfhttp://../worksheet/work_0210.pdfhttp://../worksheet/work_0209.pdf


21/22


f Able to factorise algebraic expressions.

Copy and complete this overview. If necessary, refer to the double-page opening of the chapter

and also to the Language of maths section for key words.

0 1 2 3 4 5

Low High

ALGEBRA

4a5

Four operations

2a+3a

5a3a

4k2m

4a5 =

like terms

Patterns ,,

3, 6, 9,

pattern rules

4 a=?Generalised

arithmetic sum

increase

product

quotient

subtract

Substitution 2a, when a=3

=2 3

=

5m2, when m=2

=5

2

2

=

Grouping symbols

expand

2(x+7) 2x+14

factorise

a 2a +

3 3

4k k

3 2a b

3 2

a 1

b 2

Algebraic fractions


22/22


a 9x+7x12x b 4km7mk c 3yk+8ky

d 7p+2d5p e 4w5 +2w f 8x23x5x2

g 2m+

3+

5m+

7 h 5w+

7f

2w

3f i 12d+

5

7d+

4j 2y2+3 4y2y k 7 2p+3 +5p l 6xy5 4xy2

m 3y22y+7y24y n 4a2b+5ab+3a2b2b o 5t23k7t25k


a 9 2y b 6 5k c a3d d 3 2r

e 5w3x f 2xy4y g 36a6 h 15t5t

i 20pq5q j 2y23y k 8cd24cd l 15abc20bcd

3 If a=2 and b=5, evaluate each of the following:

a ab b b3a2 c 4ab2 d a(ab)

e (2b)2

a2

f g h b(1 b)


a 5(3t4) b 7(1 +3x) c 2(4 +5k)

d 3(1 2y) e 4p(3p1) f 2m(4 3m)

g (4 9w) h (3 +2n) i 5a(a3t)

5 Expand and simplify each of the following:

a 4(2y+1) +3y b 8 +4(1 3p)

c 9g+3(1 2g) d 5(6m1) 9

e 15 4(3v5) f 4(2q+1) +3(4q+5)

g 3(1 2y) +4(4y+1) h 7r(r+1) 2r(1 3r)

i 3(2d5) 2(4 3d) j 3m2+2m(1 2m) +5m

k 2a(ab) +4b(ab) l 4e(4 3e) e(6e1)

m 3(5h2) (7 2h) n x(x+k) x(xk)

6 Factorise each of the following:

a 2m8 b x2yxy2 c 3g27 d 12x+8 16a

e 4w212w f 5k+hk2 g 8a4 h 5y25ay


a + b c d

e f g + h

i j + k 2d +5m

l 6g2k m +c

n 2h + o w


a b c d e

f g h i j

Chapter 2 Review

Ex 2-01

Ex 2-02

Ex 2-03

b 5a

b--------------- 2b2 5a

2a---------------------

Ex 2-04

Ex 2-04

Ex 2-05

Ex 2-06

h

6---

4h

6------

11k

8---------

7k

8------

7

m----

5

m----

w

4----

2w

5-------

4k3

------ k2--- 5

2d------ 4d--- x 3+

4------------ x 5+

2------------ m 1

3------------- m 2

4-------------

3y 1+

5---------------

y 4

2-----------

8

c 5+------------

5

3c 15+------------------

d

3---

3m

4-------

3g

7------

2k

3------

c 3+

4------------

c 1

5-----------

2h 1+

3---------------

h 2

5------------

2w 3+

5----------------

w 1

2-------------

Ex 2-07

5

m----

4

t---

h

k---

3

k---

14

3y------

9y

2d------

r

4---

r

5---

5

v---

2

v---

ab---

ab---

m5---- 5m

10d--------- 4x--- 12xp------

ab6

------ 5a2

------ ab8

------ 5p 2------------ 3p 6

45---------------

Topic testChapter 2
http://answers.pdf/http://answers.pdf/http://../topictest/topictest_02.pdfhttp://../topictest/topictest_02.pdfhttp://../topictest/topictest_02.pdfhttp://answers.pdf/

Algebra Chapter 02

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Transcript of Algebra Chapter 02