BEAMS KPM

Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions

TEST YOURSELF B1

1. Calculate 9  25  2. Calculate – 45   3   14 
5 27 12 7 20

3. Calculate 2 11   4. Calculate  1  4 1  
4 3 5

5. Simplify  3   m   6. Simplify n ( 5 m ) 
 k 2

7. Simplify 1 1  3 x   8. Simplify n (2a  3d ) 
6  14  2

9. Simplify  2 5x  9 y   10. Simplify x  20  1  
3  10  4  x

Curriculum Development Division 32
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions

LESSON NOTES

2.0 Division of Fractions 63
Consider the following: as 1 whole unit.

First, let’s assume this circle

Therefore, the above division can be represented visually as follows:

6 units are being divided into a group of 3
units:

63 2

This means that 6 units are being divided into a group of 3 units, or mathematically
can be written as:

6 3  2

The above division can also be interpreted as ‘how many 3’s can fit into 6’. The answer is
‘2 groups of 3 units can fit into 6 units’.

Consider now a division of a fraction by a fraction like this:

1 1. 1
28 How many is in

8
1?
2

Curriculum Development Division 33
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions

This means ‘How many is in ?

1 1
8 2

The answer is 4:

Consider now this division: How many 1 is in 3 ?
3 1. 44
44

This means ‘How many is in ?

1 3
4 4

The answer is 3: But, how do you
calculate the answer?

Curriculum Development Division 34
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions

Consider again 6  3  2.

Actually, the above division can be written as follows:

63 6 These operations are the same!
3
The reciprocal
 6 1
3 of 3 is 1 .

3

Notice that we can write the division in the multiplication form. But here, we have to
change the second number to its reciprocal.

Therefore, if we have a division of fraction by a fraction, we can do the same, that is,
we have to change the second fraction to its reciprocal and then multiply the
fractions.

Therefore, in our earlier examples, we can have:

(i) 1  1 Change the second fraction to its
28 reciprocal and change the sign  to .
 18
21 The reciprocal
8 of 1 is 8 .
2 81
4

The reciprocal of a
fraction is found by

inverting the
fraction.

Curriculum Development Division 35
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions

(ii) 3  1 Change the second fraction to its
44 reciprocal and change the sign  to .
 34
41 The reciprocal
3 of 1 is 4 .
41

The steps to divide fractions can therefore be summarized as follows:

Steps to Divide Fractions: Tips:
1. Change the second fraction to its
(+)  (+) = +
reciprocal and change the  sign to . (–) = –
(+)  –
2. Multiply the numerators together and (–)  (+) =
multiply the denominators together. (–)  (–) = +

3. Simplify the fraction (if needed).

2.1 Division of Simple Fractions
Example:

23 Change the second fraction to its reciprocal
57 and change the sign  to  .
=2  7
53 Multiply the two numerators together and
= 14 the two denominators together.
15

Curriculum Development Division 36
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions

2.2 Division of Fractions With Common Factors
Examples:

10   2 Change the second fraction to its reciprocal and
21 9 change the  sign to  .

= 10   9 Simplify by canceling out the common factors.
21 2
Multiply the two numerators together and the
=5 10   93 two denominators together.
7 21 21 Remember: (+)  (–) = (–)

=  15 Change the fraction back to a mixed number.
7

= 21
7

3 Express the fraction in division form.
5
6 Change the second fraction to its reciprocal
7 and change the  sign to  .
3  6
57 Then, simplify by canceling out the common
factors.
1
Multiply the two numerators together and the
3  7 two denominators together.
5 62

7
10

Curriculum Development Division 37
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions

1. Find 35  25 EXAMPLES
12 6
Change the second fraction to its reciprocal
Solution : 35  25 and change the  sign to .
12 6
Then, simplify by canceling out the common
= 7 35  6 1 factors.
2 12 25
Multiply the two numerators together and the
=7 5 two denominators together.
10
Change the second fraction to its reciprocal
2. Simplify – 2  5x and change the  sign to .
x4
Multiply the two numerators together and the two
Solution : – 2  4 denominators together.
x 5x
Express the fraction in division form.
= –8 Change the second fraction to its reciprocal
5x2
and change  to  .
y Multiply the two numerators together and the two
3. Simplify x
denominators together.
2 Remember: (+)  (–) = (–)
Solution :

Method I y   2
x
 y  1
x2
 y
2x

Curriculum Development Division 38
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions

Method II The given fraction.

y The numerator is also
x a fraction with
2 denominator x

y MMuullttiippllyy tthhee nnuummeerraattoorr aanndd tthhee ddeennoommiinnaattoorr ooff the
= x x the given fractiognivweinthfxraction by x.

2 x

y x
=x

2 x
= y

2x

(1  1 )
r
4. Simplify
5

Solution: 1
r is the denominator of .
(1  1 ) r
r r r
Multiply the given fraction with r .
5
r
(1  1 )
=r Note that:
(1  1)  r  r 1
5
= r 1 r

5r

Curriculum Development Division 39
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions

TEST YOURSELF B2

1. Calculate  3   21 2. Calculate 5   7  5
72 9 8 16

3. Simplify 8   4 y 4. Simplify 16
y3 2
k
5. Simplify 2
5 x 6. Simplify  4m   2m2
n 3n
3

4 8. Simplify x
y 1 1 1
7. Simplify x
8

Curriculum Development Division 40
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions

3(1  1 ) 5 1
4 x
9. Calculate 10. Simplify
5
y

 x 1  4 1
11. Simplify 9 p
2 12. Simplify
1 1
3 5

Curriculum Development Division 41
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions

ANSWERS

TEST YOURSELF A: 2. 1 3. 5
1. 3 2 14
7
4. 1 5. 38 or 1 3 6.  3
4 35 35 14
7. 67 or 5 2
13 13 8. 73 or 1 28 9. 3
10. 6 45 45 s
w 1
13. 2b  4a 11. 5
ab 2a 12.
q 5p 3f
16. 3 p  3
2 14. 15. m  n
1 pq
18. 2x 1
19. 17. 16x  17 y x
x(x  1) 10
21. 8x  y
22. 7n  4 20. 2 2
9n 2
r2 1  p2 6
7n  4n2  6 23. 24. 2 p2
25. 10n2 27. n  5
28. n  3 3r
26. 1  m 5n
3n 30. 4 p  3
m
29. n 10 3m

8n 2

Curriculum Development Division 42
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions

TEST YOURSELF B1:

1. 5 or 1 2 2.  9 or 11 3. 11 or 5 1
33 88 22

4.  7 or 1 2 5. 3m 6. 5mn
55 k 2

7. x 8. na  3 nd 9.  10 x  3 y
4 2 35

10. 5x  1
4

TEST YOURSELF B2: 2.  14 or 1 5 3.  6
99 y2
1. 2
49 5. 6 6. 6
5 x m
4. 8k
1 8. x2 9. 9
x 1 20
7. 2( y 1)
5x 1 11. 13x 12.  5
6 4p
10.
xy

Curriculum Development Division 43
Ministry of Education Malaysia

Basic Essential
Additional Mathematics Skills

UNIT 3

ALGEBRAIC EXPRESSIONS
AND

UAnLitG1E: BRAIC FORMULAE
Negative Numbers

Curriculum Development Division
Ministry of Education Malaysia

TABLE OF CONTENTS 1
2
Module Overview 10
Part A: Performing Operations on Algebraic Expressions 15
Part B: Expansion of Algebraic Expressions 23
Part C: Factorisation of Algebraic Expressions and Quadratic Expressions
Part D: Changing the Subject of a Formula 31
Activities 33
37
Crossword Puzzle 38
Riddles
Further Exploration
Answers

Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae

MODULE OVERVIEW

1. The aim of this module is to reinforce pupils’ understanding of the concepts and skills
in Algebraic Expressions, Quadratic Expressions and Algebraic Formulae.

2. The concepts and skills in Algebraic Expressions, Quadratic Expressions and
Algebraic Formulae are required in almost every topic in Additional Mathematics,
especially when dealing with solving simultaneous equations, simplifying
expressions, factorising and changing the subject of a formula.

3. It is hoped that this module will provide a solid foundation for studies of Additional
Mathematics topics such as:
 Functions
 Quadratic Equations and Quadratic Functions
 Simultaneous Equations
 Indices and Logarithms
 Progressions
 Differentiation
 Integration

4. This module consists of four parts and each part deals with specific skills. This format
provides the teacher with the freedom to choose any parts that is relevant to the skills
to be reinforced.

Curriculum Development Division 1
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae

PART A:

PERFORMING OPERATIONS ON
ALGEBRAIC EXPRESSIONS

LEARNING OBJECTIVES

Upon completion of Part A, pupils will be able to perform operations on algebraic
expressions.

TEACHING AND LEARNING STRATEGIES

Pupils who face problem in performing operations on algebraic expressions might have
difficulties learning the following topics:

 Simultaneous Equations - Pupils need to be skilful in simplifying the algebraic
expressions in order to solve two simultaneous equations.

 Functions - Simplifying algebraic expressions is essential in finding composite
functions.

 Coordinate Geometry - When finding the equation of locus which involves
distance formula, the techniques of simplifying algebraic expressions are required.

 Differentiation - While performing differentiation of polynomial functions, skills
in simplifying algebraic expressions are needed.

Strategy:

1. Teacher reinforces the related terminologies such as: unknowns, algebraic terms,
like terms, unlike terms, algebraic expressions, etc.

2. Teacher explains and shows examples of algebraic expressions such as:
8k, 3p + 2, 4x – (2y + 3xy)

3. Referring to the “Lesson Notes” and “Examples” given, teacher explains how to
perform addition, subtraction, multiplication and division on algebraic expressions.

4. Teacher emphasises on the rules of simplifying algebraic expressions.

Curriculum Development Division 2
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae

LESSON NOTES

PART A:
PERFORMING BASIC ARITHMETIC OPERATIONS ON ALGEBRAIC EXPRESSIONS

1. An algebraic expression is a mathematical term or a sum or difference of mathematical
terms that may use numbers, unknowns, or both.

Examples of algebraic expressions: 2r, 3x + 2y, 6x2 +7x + 10, 8c + 3a – n2, 3

g

2. An unknown is a symbol that represents a number. We normally use letters such as n, t, or
x for unknowns.

3. The basic unit of an algebraic expression is a term. In general, a term is either a number
or a product of a number and one or more unknowns. The numerical part of the term, is
known as the coefficient.

Coefficient Unknowns

6 xy

Examples: Algebraic expression with one term: 2r, 3

Algebraic expression with two terms: g
Algebraic expression with three terms:
3x + 2y, 6s – 7t

6x2 +7x + 10, 8c + 3a – n2

4. Like terms are terms with the same unknowns and the same powers.

Examples: 3ab, –5ab are like terms.

3x2, 2 x2 are like terms.
5

5. Unlike terms are terms with different unknowns or different powers.
Examples: 1.5m, 9k, 3xy, 2x2y are all unlike terms.

Curriculum Development Division 3
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae

6. An algebraic expression with like terms can be simplified by adding or subtracting the
coefficients of the unknown in algebraic terms.

7. To simplify an algebraic expression with like terms and unlike terms, group the like terms
first, and then simplify them.

8. An algebraic expression with unlike terms cannot be simplified.
9. Algebraic fractions are fractions involving algebraic terms or expressions.

Examples: 3m , 2 , 4r 2 g , x2  y2 .
15 6h 2rg  g 2 x2  2xy  y2

10. To simplify an algebraic fraction, identify the common factor of both the numerator and the
denominator. Then, simplify it by elimination.

Curriculum Development Division 4
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae

EXAMPLES

Simplify the following algebraic expressions and algebraic fractions:

(a) 5x – (3x – 4x) (e) s  t
(b) –3r –9s + 6r + 7s 46

(f ) 5x  3y
6 2z

4r 2 g (g) e  2g
(c) 2rg  g 2 f

(d) 3  4 3x  1
pq (h) 2

3x

Solutions: Algebraic expression with like terms can be simplified by
(a) 5x – (3x – 4x) adding or subtracting the coefficients of the unknown.

= 5x – (– x) Perform the operation in the bracket.
= 5x + x
= 6x

(b) –3r –9s + 6r + 7s Arrange the algebraic terms according to the like terms.
= –3r + 6r –9s + 7s Unlike terms can.not be simplified.
= 3r – 2s
Leave the answer in the simplest form as shown.
4r 2 g
(c) Simplify by canceling out the common factor and the
same unknowns in both the numerator and the
2rg  g 2 denominator.
 4r 2g 1

g(2r  g)

1

 4r 2
2r  g

Curriculum Development Division 5
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae

(d) 3  4 The LCM of p and q is pq.
pq The LCM of 4 and 6 is 12.

 3q  4 p
pq pq

 3q  4 p
pq

(e) s  t
46
 3s  2t
43 62
 3s  2t
12

(f ) 5x  1  5x  y Simplify by canceling out the common
factor, then multiply the numerators
3y together and followed by the
denominators.
2 6 2z 22z
Change division to multiplication of the
 5xy reciprocal of 2g.
4z
Equate the denominator.
(g) e  2g  e  1
f f 2g
e
2 fg

3x  1 3x(2)  1
(h) 2  2 2

3x 3x
6x 1

2
3x

 6x 1 1
2 3x

 6x 1
6x

Curriculum Development Division 6
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae

ALTERNATIVE METHOD

Simplify the following algebraic fractions:

3x  1 3x  1  The denominator of 1 is 2 . Therefore,
(a) 2 =  2  2 2
2
3x 3x 2
multiply the algebraic fraction by .
3x(2)  1 (2)
=2 2

3x(2) Each of the terms in the numerator and
denominator of the algebraic fraction is
multiplied by 2.

= 6x 1
6x

(b) 32  3  2 x 3
x = x 
The denominator of is x. Therefore,
5 5x
x
3 (x)  2(x) x
x
multiply the algebraic fraction by .
5( x)
x

Each of the terms in the numerator and
denominator is multiplied by x.

 3 2x
5x

8   3  8   3  2x 3
 2x    2x
(c)  The denominator of is 2x. Therefore,
2 2 2x
2x
8(2x)   3 (2x) 2x
  2x 
multiply the algebraic fraction by .
2(2x)
2x

Each of the terms in the numerator and
denominator is multiplied by 2x.

.

 16x  3
4x

Curriculum Development Division 7
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae

(d) 3  3  7 The denominator of 8 x is 7.

 8  x   4  8  x   4 7 7

7 7 Therefore, multiply the algebraic

 3(7) 7
 8  x (7)  4(7)
7 fraction by .

7

 21 Each of the terms in the numerator
8  x  28 and denominator is multiplied by 7.

 21 Simplify the denominator.
36  x

Curriculum Development Division 8
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae

TEST YOURSELF A

Simplify the following algebraic expressions: 2. − 4m + 5n + 2m – 9n
1. 2a –3b + 7a – 2b

3. 8k – ( 4k – 2k ) 4. 6p – ( 8p – 4p )

5. 3  1 6. 4h  2k
y 5x 35

7. 4a  3b 8. 4c  d  8
7 2c 2 3c  d

9. xy  yz 10. u  uv
z vw 2w
 4   2
11. 2
 5   6 12.  x 
x  4   5
x

Curriculum Development Division 9
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae

PART B:

EXPANSION OF ALGEBRAIC
EXPRESSIONS

LEARNING OBJECTIVE
Upon completion of Part B, pupils will be able to expand algebraic
expressions.

TEACHING AND LEARNING STRATEGIES

Pupils who face problem in expanding algebraic expressions might have
difficulties in learning of the following topics:

 Simultaneous Equations – pupils need to be skilful in expanding the
algebraic expressions in order to solve two simultaneous equations.

 Functions – Expanding algebraic expressions is essential when finding
composite function.

 Coordinate Geometry – when finding the equation of locus which
involves distance formula, the techniques of expansion are applied.

Strategy:
Pupils must revise the basic skills involving expanding algebraic expressions.

Curriculum Development Division 10
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae

LESSON NOTES

PART B:
EXPANSION OF ALGEBRAIC EXPRESSIONS

1. Expansion is the result of multiplying an algebraic expression by a term or another
algebraic expression.

2. An algebraic expression in a single bracket is expanded by multiplying each term in the
bracket with another term outside the bracket.

3(2b – 6c – 3) = 6b – 18c – 9

3. Algebraic expressions involving two brackets can be expanded by multiplying each term of
algebraic expression in the first bracket with every term in the second bracket.

(2a + 3b)(6a – 5b) = 12a2 – 10ab + 18ab – 15b2
= 12a2 + 8ab – 15b2

4. Useful expansion tips:
(i) (a + b)2 = a2 + 2ab + b2
(ii) (a – b)2 = a2 – 2ab + b2
(iii) (a – b)(a + b) = (a + b)(a – b)
= a2 – b2

Curriculum Development Division 11
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae

EXAMPLES

Expand each of the following algebraic expressions:

(a) 2(x + 3y) (d) (a  3)2

(b) – 3a (6b + 5 – 4c) (e)  32k  52

(c) 2 9 y  12 (f ) ( p  2)( p  5)

3

Solutions:

(a) 2 (x + 3y) When expanding a bracket, each term
= 2x + 6y within the bracket is multiplied by the term

outside the bracket.

(b) –3a (6b + 5 – 4c) When expanding a bracket, each term
= –18ab – 15a + 12ac within the bracket is multiplied by the term

outside the bracket.

(c) 2 9 y  12 Simplify by canceling out the common
factor, then multiply the numerators
3
together and followed by the denominators.
= 2  3  2  4
When expanding two brackets, each term
9y 12 within the first bracket is multiplied by
13 13 every term within the second bracket.

= 6y + 8

(d) (a  3)2

= (a + 3) (a + 3)

= a2 + 3a + 3a + 9
= a2 + 6a + 9

Curriculum Development Division 12
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae

(e)  32k  52 When expanding two brackets, each term
within the first bracket is multiplied by
= –3(2k + 5) (2k + 5) every term within the second bracket.
= –3(4k2 + 20k + 25)
= –12k2 – 60k – 75

(f ) ( p  2) (q  5) When expanding two brackets, each term
= pq – 5p + 2q – 10 within the first bracket is multiplied by
every term within the second bracket.
ALTERNATIVE METHOD

Expanding two brackets

(a) (a + 3) (a + 3) When expanding two
brackets, write down the
= a2 + 3a + 3a + 9 product of expansion and
= a2 + 6a + 9
then, simplify the like
(b) (2p + 3q) (6p – 5q) (c) (4x – 3teyr)m(6sx. – 5y)

= 12p2 – 10 pq + 18 pq – 15q2 – 18 xy
= 12p2 + 8 pq – 15q2 – 20 xy
– 38 xy

= 24x2 – 38 xy + 15y2

Curriculum Development Division 13
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae

TEST YOURSELF B

Simplify the following expressions and give your answers in the simplest form.

1.  4 2n  3  2. 1 6q 1
 4
2

3.  6x2x  3y 4. 2a  b  2(a  b)

5. 2( p  3)  ( p  6) 6. 1 6x  y   x  2 y 

3  3

7. e 12  2e 1 8. m  n2  m2m  n

9.  f  g f  g  g2 f  g 10. h  ih  i 2ih  3i

Curriculum Development Division 14
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae

PART C:

FACTORISATION OF
ALGEBRAIC EXPRESSIONS AND

QUADRATIC EXPRESSIONS

LEARNING OBJECTIVE

Upon completion of Part C, pupils will be able to factorise algebraic expressions
and quadratic expressions.

TEACHING AND LEARNING STRATEGIES

Some pupils may face problem in factorising the algebraic expressions. For
example, in the Differentiation topic which involves differentiation using the
combination of Product Rule and Chain Rule or the combination of Quotient
Rule and Chain Rule, pupils need to simplify the answers using factorisation.

Examples:
1. y  2x3 (7x  5)4

 dy  2x3[28(7x  5)3 ] (7x  5)4 (6x2 )
dx
 2x2 (7x  5)3 (49x 15)

2. y  (3  x)3
7  2x

 dy  (7  2x)[3(3  x)2 ]  (3  x)3 (2)
dx (7  2x)2
 (3  x)2 (4x 15)
(7  2x)2

Strategy
1. Pupils revise the techniques of factorisation.

Curriculum Development Division 15
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae

LESSON NOTES

PART C:
FACTORISATION OF
ALGEBRAIC EXPRESSIONS AND QUADRATIC EXPRESSIONS

1. Factorisation is the process of finding the factors of the terms in an algebraic expression. It
is the reverse process of expansion.

2. Here are the methods used to factorise algebraic expressions:
(i) Express an algebraic expression as a product of the Highest Common Factor (HCF) of
its terms and another algebraic expression.
ab – bc = b(a – c)
(ii) Express an algebraic expression with three algebraic terms as a complete square of two
algebraic terms.
a2 + 2ab + b2 = (a + b)2
a2 – 2ab + b2 = (a – b)2
(iii) Express an algebraic expression with four algebraic terms as a product of two algebraic
expressions.
ab + ac + bd + cd = a(b + c) + d(b + c)
= (a + d)(b + c)
(iv) Express an algebraic expression in the form of difference of two squares as a product of
two algebraic expressions.
a2 – b2 = (a + b)(a – b)

3. Quadratic expressions are expressions which fulfill the following characteristics:
(i) have only one unknown; and
(ii) the highest power of the unknown is 2.

4. Quadratic expressions can be factorised using the methods in 2(i) and 2(ii).
5. The Cross Method can be used to factorise algebraic expression in the general form of

ax2 + bx + c, where a, b, c are constants and a ≠ 0, b ≠ 0, c ≠ 0.

Curriculum Development Division 16
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae

EXAMPLES

(a) Factorising the Common Factors

i) mn + m = m (n +1) Factorise the common factor m.
ii) 3mp + pq = p (3m + q)
iii) 2mn – 6n = 2n (m – 3) .
Factorise the common factor p.

.
Factorise the common factor 2n.

.

(b) Factorising Algebraic Expressions with Four Terms

i) vy + wy + vz + wz Factorise the first and the second terms
= y (v + w) + z (v + w) with the common factor y, then factorise
= (v + w)(y + z)
the third and fourth terms with the
common factor z.

.
(v + w) is the common factor.

ii) 21bm – 7bs + 6cm – 2cs Factorise the first and the second terms with
= 7b(3m – s) + 2c(3m – s) common factor 7b, then factorise the third
= (3m – s)(7b + 2c) and fourth terms with common factor 2c.

(3m – s) is the common factor.

Curriculum Development Division 17
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae

(c) Factorising the Algebraic Expressions by Using Difference of Two Squares
a2 – b2 = (a + b)(a – b)

i) x2 – 16 = x2 – 42
= (x + 4)(x – 4)

ii) 4x2 – 25 = (2x)2 – 52
= (2x + 5)(2x – 5)

(d) Factorising the Expressions by Using the Cross Method

i) x2 – 5x + 6 The summation of the cross
multiplication products should
x 3 equal to the middle term of the
x 2
 3x  2x  5x quadratic expression in the
general form.

x2 – 5x + 6 = (x – 3) (x – 2)

ii) 3x2 + 4x – 4 The summation of the cross
multiplication products should
3x  2 equal to the middle term of the
x 2
 2x  6x  4x quadratic expression in the
general form.

3x2 + 4x – 4 = (3x – 2) (x + 2)

Curriculum Development Division 18
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae

ALTERNATIVE METHOD

Factorise the following quadratic expressions: REMEMBER!!!

i) x 2 – 5x + 6 An algebraic expression can
be represented in the general
a=+1 b= –5 c =+6 form of ax2 + bx + c, where

a, b, c are constants and
a ≠ 0, b ≠ 0, c ≠ 0.

+1  (+ 6) = + 6 ac b –2  (–3) = +6
+6 –5 –2 + (–3) = –5
–2 –3

(x – 2) (x – 3)
 x2  5x  6  (x  2)(x  3)

ii) x 2 – 5x – 6

a=+1 b= –5 c = –6

+1  (–6) = –6

ac b +1  (–6) = –6
–6 –5 +1 – 6 = –5
+1 – 6

(x + 1) (x– 6)
 x2  5x  6  (x 1)(x  6)

Curriculum Development Division 19
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae

(iii) 2x2 – 11x + 5

a=+2 b = –11 c =+5

(+2)  (+5) = +10 ac b
+ 10 –11
–1  (–10) = +10
–1 – 10 –1 + (–10) = –11
 1  10
The coefficient of x2 is 2,
22 divide each number by 2.
 1 5

2

The coefficient of x2 is 2,

(2x – 1) (x – 5) multiply by 2:
 2x2 11x  5  (2x 1)(x  5)
x  1 x  5
2

 2x  1 x  5
2

 2x 1)(x  5

TEST YOURSELF C

(iv) 3x2 + 4x – 4

a =+ 3 b=+ 4 c = –4

ac b –2 + 6 = 4
– 12 + 4
3  (– 4) = –12 – 2 +6
2 6
The coefficient of x2 is 3, divide each
33 number by 3.

The coefficient of x2 is 3, multiply by 3:

 2 2 x  2 x  2
3 3

 3x  2 x  2
3

(3x – 2) (x + 2)  3x  2)(x  2
 3x2  4x  4 (3x  2)(x  2)
20
Curriculum Development Division
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae

TEST YOURSELF C

Factorise the following quadratic expressions completely.

1. 3p 2 – 15 2. 2x 2 – 6

3. x 2 – 4x 4. 5m 2 + 12m

5. pq – 2p 6. 7m + 14mn

7. k2 –144 8. 4p 2 – 1

9. 2x 2 – 18 10. 9m2 – 169

Curriculum Development Division 21
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae

11. 2x 2 + x – 10 12. 3x 2 + 2x – 8

13. 3p 2 – 5p – 12 14. 4p2 – 3p – 1

15. 2x 2 – 3x – 5 16. 4x 2 – 12x + 5

17. 5p 2 + p – 6 18. 2x 2 – 11x + 12

19. 3p + k + 9pr + 3kr 20. 4c2 – 2ct – 6cw + 3tw

Curriculum Development Division 22
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae

PART D:

CHANGING THE SUBJECT
OF A FORMULA

LEARNING OBJECTIVE

Upon completion of this module, pupils will be able to change the subject of
a formula.

TEACHING AND LEARNING STRATEGIES

If pupils have difficulties in changing the subject of a formula, they probably
face problems in the following topics:

 Functions – Changing the subject of the formula is essential in finding
the inverse function.

 Circular Measure – Changing the subject of the formula is needed to

find the r or  from the formulae s = r or A  1 r 2 .

2
 Simultaneous Equations – Changing the subject of the formula is the

first step of solving simultaneous equations.

Strategy:

1. Teacher gives examples of formulae and asks pupils to indicate the subject

of each of the formula.

Examples: y=x–2
A  1 bh
y, A and V are the
2 subjects of the
V  r 2h formulae.

Curriculum Development Division 23
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae

LESSON NOTES

PART D:
CHANGING THE SUBJECT OF A FORMULA

1. An algebraic formula is an equation which connects a few unknowns with an equal
sign.

Examples: A  1 bh
2

V  r 2h

2. The subject of a formula is a single unknown with a power of one and a coefficient
of one, expressed in terms of other unknowns.

Examples: A  1 bh A is the subject of the formula because it is
2 expressed in terms of other unknowns.

a2 = b2 + c2 a2 is not the subject of the formula
because the power ≠ 1

T  1 Tr2h T is not the subject of the formula
2 because it is found on both sides of the

equation.

3. A formula can be rearranged to change the subject of the formula. Here are the
suggested steps that can be used to change the subject of the formula:

(i) Fraction : Get rid of fraction by multiplying each term in the formula with
the denominator of the fraction.

(ii) Brackets : Expand the terms in the bracket.

(iii) Group : Group all the like terms on the left or right side of the formula.

(iv) Factorise : Factorise the terms with common factor.

(v) Solve : Make the coefficient and the power of the subject equal to one.

Curriculum Development Division 24
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae

EXAMPLES

Steps to Change the Subject of a Formula

(i) Fraction
(ii) Brackets
(iii) Group
(iv) Factorise
(v) Solve

1. Given that 2x + y = 2, express x in terms of y.

Solution: No fraction and brackets.

2x + y = 2 Group:
2x = 2 – y Retain the x term on the left hand side of the

x= 2 y equation by grouping all the y term to the
2 right hand side of the equation.

Solve:
Divide both sides of the equation by 2 to

make the coefficient of x equal to 1.

2. Given that 3x  y  5y , express x in terms of y.
2

Solution:

3x  y  5y Fraction:
2 Multiply both sides of the equation by 2.

3x + y = 10y Group:
3x = 10y – y Retain the x term on the left hand side of the
3x = 9y
x = 9y equation by grouping all the y term to the
3 right hand side of the equation.
x = 3y
Solve:
Divide both sides of the equation by 3 to

make the coefficient of x equal to 1.

Curriculum Development Division 25
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae

3. Given that x  2 y , express x in terms of y.
Solution:

x  2y Solve:
x = (2y) 2 Square both sides of the equation to make the

power of x equal to 1.

x = 4y 2

4. Given that x  p , express x in terms of p.
3

Solution:

xp Fraction:
3 Multiply both sides of the equation by 3.

x  3p Solve:
x  (3 p)2 Square both sides of the equation to make
x  9p2
the power of x equal to1.

5. Given that 3 x  2  x  y , express x in terms of y.

Solution: Group:
Group the like terms
3 x2 xy
3 x x  y2 Simplify the terms.

2 x  y2 Solve:
x  y2 Divide both sides of the equation by 2 to
2
x   y  2 2 make the coefficient of x equal to 1.
2
Solve:
Square both sides of equation to make the

power of x equal to 1.

Curriculum Development Division 26
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae

6. Given that 11x – 2(1 – y) = 2xp , express x in terms of y and p.
4

Solution: Fraction:
11x – 2 (1 – y) = 2xp Multiply both sides of the equation
4
11x – 8(1 – y) = 8xp by 4.
11x – 8 + 8y = 8xp
11x – 8xp = 8 – 8y Bracket:
Expand the bracket.

Group:
Group the like terms.

x(11 – 8p) = 8 – 8y Factorise:
x = 88y Factorise the x term.
11  8 p
Solve:
Divide both sides by (11 – 8p) to
make the coefficient of x equal to 1.

7. Given that 2 p  3x = 1 – p , express p in terms of x and n.
5n

Solution: Fraction:
2 p  3x = 1 – p Multiply both sides of the equation by
5n
2p – 3x = 5n – 5pn 5n.
2p + 5pn = 5n + 3x
p(2 + 5n) = 5n + 3x Group:
p = 5n  3x Group the like p terms.
2  5n
Factorise:
Factorise the p terms.

Solve:
Divide both sides of the equation by
(2 + 5n) to make the coefficient of p

equal to 1.

Curriculum Development Division 27
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae

1. Express x in terms of y. TEST YOURSELF D
a) x  y  2  0
b) 2x  y  3  0

c) 2 y  x  1 d) 1 x  y  2

2

e) 3x  y  5 f) 3y  x  4

Curriculum Development Division 28
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae

2. Express x in terms of y. b) 2 y  x
a) y  x

c) 2 y  x d) y  1  3 x
3

e) 3 x  y  x  1 f) x  1  y

Curriculum Development Division 29
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae

3. Change the subject of the following formulae:

a) Given that x  a  2 , express x in terms b) Given that y  1  x , express x in terms

xa 1 x

of a . of y .

c) Given that 1  1  1 , express u in d) Given that 2 p  q  3 , express p in

f uv 2pq 4

terms of v and f . terms of q.

e) Given that p  3m  2mn , express m in f) Given that A  B C 1 , express C in
terms of n and p .
C

terms of A and B .

g) Given that 2 y  x  2 y , express y in h) Given that T  2 l
x , express g in
g
terms of x.
terms of T and l.

Curriculum Development Division 30
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae

ACTIVITIES

CROSSWORD PUZZLE

HORIZONTAL .
1) – 4p, 10q and 7r are called algebraic

3) An algebraic term is the of unknowns and numbers.

4) 4m and 8m are called terms.

5) V  r 2h , then V is the of the formula.

7) An can be represented by a letter.

10) x2  3x  2  x 1x  2.

Curriculum Development Division 31
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae

VERTICAL

2) An algebraic consists of two or more algebraic terms combined by

addition or subtraction or both.

6) 2x 1x  2  2x2  5x  2 .

8) terms are terms with different unknowns.

9) The number attached in front of an unknown is called .

Curriculum Development Division 32
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae

RIDDLES

RIDDLE 1
1. You are given 9 multiple-choice questions.
2. For each of the questions, choose the correct answer and fill the alphabet in the box
below.
3. Rearrange the alphabets to form a word.
4. What is the word?

123456789

2 1 O) 1
1. Calculate 5 . N) 11

3 15
D) 1
W)  9x 16y
5 X) 9x  2y
W) 11

3

2. Simplify  3x  9y  6x  7y .
F) 3x  2y
E) 3x  2y

3. Simplify p  q . A) 2 p  3q
32 6

L) 2 p  3q R) 3 p  2q
6 6

N) 3q  2 p
6

Curriculum Development Division 33
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae

4. Expand 2(x  4)  (x  7) . D) x 15
C) 3x 15
A) x 1
U) 3x 1

5. Expand  3a(2b  5c) . C) 6ab 15ac
R) 6ab 15ac
S )  6ab 15ac
T)  6ab 15ac

6. Factorise x2  25 . T) (x  5)(x  5)
E) (x  5)(x  5) C) (x  25)(x  25)
I) (x  5)(x  5)
E) q( p  4)
7. Factorise pq  4q . S) q( p  4)
D) pq(1 4q)
T) p(q  4)

8. Factorise x2  8x 12 . W) (x  2)(x  6)
I ) (x  2)(x  6) C) (x  4)(x  3)
F) (x  4)(x  3)

9. Given that 3x  y  4 , express x in terms of y.
2x

L) x   y C) x  y
5 5

T) x  y N) x  8  y
11 3

Curriculum Development Division 34
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae

RIDDLE 2

1. You are given 9 multiple-choice questions.
2. For each of the questions, choose the correct answer and fill the alphabet in the box

below.
3. Rearrange the alphabets to form a word.
4. What is the word?

123456789

5 1 O) 5  x
1. Calculate x . 3x

3 N) 3
x5
A) 5  x
3 R) 4q
15 pr
I ) 3x
x5 B) 3 pq
5r
2. Simplify 3p  q .
4 5r

F) 15 pr
4q

W) 3 pq
20r

3. Simplify x  xy .
yz 2z

N) 2 D) x2
y2 2z2

L) x x2
2z2 I)

z2

4. Solve x  y2  x(3x  y). D) 2x 2  y 2  xy
N) 2x 2  y 2  xy
E)  2x2  y 2  xy
I ) x 2  y 2  3x 2  xy

Curriculum Development Division 35
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae

5. Expand  p  52 .

I) p 2  25 N) p 2  25
D) p 2 10 p  25 L) p 2  10 p  25

6. Factorise 2 y 2  7 y 15 . D) (2y  3)(y  5)
L) ( y  3)(2y  5)
F) (2y  3)(y  5)
W) (2y  3)(y  5)

7. Factorise 2 p2 11 p  5 . B) (2 p 1)( p  5)
W) ( p 1)(2 p  5)
R) (2 p 1)( p  5)
F) ( p 1)( p  5)

8. Given that B (C 1)  A , express C in terms of A and B.
C

L) C  B R) C  1
BA BA

C) C  AB N) C  AB
BA BA

9. Given that 5 x  y  x  2 , express x in terms of y.

O) x  y 2  4 B) x  y 2  4
16 24

I ) x   y 12 U) x   y  2 2
2 4

Curriculum Development Division 36
Ministry of Education Malaysia