NOTE MATH FORM 3 - ALGEBRAIC

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Wednesday, April 12, 2023

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Expand a single pair of brackets

Example 1: Expand w (x + y)

w ( x + y )

= w x x + w x y

= wx + wy

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Example 2: Expand 3 (a - c)

3 ( a - c )

= 3 x a - 3 x c

= 3a - 3c

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Example 3: Expand 7g (m + 2n)

7g ( m + 2n )

= 7g x m + 7g x 2n

= 7gm + 14gn

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Example 4: Expand 4e ( r + 2s – 8t)

4e ( r + 2s - 8t )

= 4e x r + 4e x 2s

= 4er + 8es

- 4e x 8t

- 32et

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ALGEBRAIC EXPRESSIONS

EXPAND DOUBLE PAIRS OF BRACKETS

HJ. AIRIL AHMAD

AIRIL AHMAD


Expand double pairs of brackets

Example 1: Expand (w + v)(x + y)

( w + v ) ( x + y )

= w x x + w x y + v x x + v x y

= wx + wy + vx + vy

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Example 2: Expand (2e + f)(c + d)

( 2e + f ) ( c + d )

= 2e x c + 2e x d + f x c + f x d

= 2ce + 2de + cf + df

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Example 3: Expand (2m + n)(p - s)

( 2m + n ) ( p - s )

= 2m x p + 2m x (-s) + n x p + n x (-s)

= 2mp - 2ms + np - ns

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HJ. AIRIL AHMAD

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Expand double pairs of bracketsExample 1: Expand (r + 4)(r + 3)

( r + 4 ) ( r + 3 )

= r x r + r x 3 + 4 x r + 4 x 3

= r2 + 3r + 4r + 12

= r2 + 7r + 12

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Expand double pairs of bracketsExample 2: Expand (2e + f)(e + 2f)

( 2e + f ) ( e + 2f )

= 2e x e + 2e x 2f + f x e + f x 2f

= 2e2 + 4ef + ef + 2f2

= 2e2 + 5ef + 2f2

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EXERCISE

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Expand double pairs of bracketsExample 3: Expand (w + 4)(w - 3)

( w + 4 ) ( w - 3 )

= w x w + w x (-3) + 4 x w + 4 x (-3)

= w2 - 3w + 4w - 12

= w2 + w - 12

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In general , (a+b)2= a2 + 2ab + b2 or (a-b)2= a2 - 2ab + b2

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Expand same pairs of brackets

Example 1: Expand (r + 4)2

( r + 4 ) ( r + 4 )

= r x r + r x 4 + 4 x r + 4 x 4

= r2 + 4r + 4r + 16

= r2 + 8r + 16

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Example 2: Expand (2e + f)2

( 2e + f ) ( 2e + f )

= 2e x 2e + 2e x f + f x 2e + f x f

= 4e2 + 2ef + 2ef + f2

= 4e2 + 4ef + f2

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Example 3: Expand (w - 4)2

( w - 4 ) ( w - 4 )

= w x w + w x (-4) + (-4) x w + -(4) x (-4)

= w2 - 4w - 4w + 16

= w2 -8w + 16

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State the factors of an algebraic term

Example 1: 2m

1,

Example 2: 4g

2, m, 2m

1, 2, g, 2g, 4g4,

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State the factors of an algebraic term

Example 3: 12y

1,

Example 4: 15k

2, 6y, 12y

1, 3, 3k, 5k, 15k5,

3, 4y,4, 3y,12, y,6, 2y,

15, k,

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Exercise

Write all the factors of the following algebraic terms.

1. 5 j

2. 14 p

3. 5 n2

4. abc

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CONCEPT OF FACTORISATION OF

ALGEBRAIC EXPRESSIONS

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Highest Common Factor (HCF)

Find the HCF of the following algebraic terms.

6h , 4gh 3h , 2gh

3 , 2g

HCF = 2 x h = 2h

Example 1 : 6h & 4gh

2pq , 4q pq , 2q

p , 2

HCF = 2 x q = 2q

Example 2 : 2pq & 4q

h 2 2

q

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Highest Common Factor (HCF)

Find the HCF of the following algebraic terms.

14v2 , 8v 7v2 , 4v

7v , 4

HCF = 2 x v = 2v

Example 3 : 14v2 & 8v

5cd2 , 10cd cd2 , 2cd

d , 2

HCF = 5 x cd = 5cd

Example 4 : 5cd2 & 10cd

2 v

5 cd

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EXERCISE

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FACTORISE ALGEBRAIC

EXPRESSIONS

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Factorise the following algebraic term

Example 1: 2pq + 4qr

2pq + 4qr pq + 2qr p + 2r (p + 2r)

2q2q

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Example 2: mn – mn2

mn - mn2

n - n2

1 - n (1 - n)

mnmn

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Example 3: 3ab + 6bc - 9b2

3ab + 6bc - 9b2

ab + 2bc - 3b2 a + 2c - 3b( a + 2c - 3b)

3b3b

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EXERCISE

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FACTORISE ALGEBRAIC

EXPRESSIONS

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aa( ) ( )

Factorise ab + ac +bd + cd

Example 9: ag + 3a + 2g + 6

ag + 3a + 2g + 62+ 2(g + 3) (g + 3)

(g + 3) (g + 3)(a + 2 )

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nn( ) ( )

Factorise ab + ac +bd + cd

Example 10: mn + 4n + 2m + 8

mn + 4n + 2m + 82+ 2(m + 4) (m + 4)

(m + 4) (m + 4)(n + 2 )

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EXERCISE

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Exercise

Factorise the following algebraic expressions:

1. ac + ad + bc + bd2. ax + ay + bx + by3. gh + 2h + 3g + 64. a2 + ab + 2a + 2b5. de + d2 + ef + df

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( a + b )(a

Factorise a2- b2

Example 11: a2 – b2

a2 – b2- b )

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( m + n )(m

Factorise a2- b2

Example 12: m2 – n2

m2 – n2- n )

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( 4m + 5n )(4m

Factorise a2- b2

Example 13: 16m2 – 25n2

16m2 – 25n2

- 5n ) 42m2 – 52n2

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EXERCISE

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FACTORISE ALGEBRAIC

EXPRESSIONS

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++4)(y

+

4

Factorise a2 + 2ab + b2

16 = 1 x 16

16 = 2 x 8

16 = 4 x 4

8 = 4 + 4

yy 41st Step:

2nd Step:

Example 1: y2 + 8y + 16

+4y

+4y+(y + 4)

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+

++2)(y

+

2

14 = 1 x 14

14 = 2 x 7

9 = 2 + 7

yy 71st Step:

2nd Step:

Example 2: y2 + 9y + 14

+2y

+7y

(y + 7)

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Exercise:• Page 135• Ex.6.2C• 1 - 8

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Factorise completely (Factorise a2 + 2ab + b2)

1. m2 + 12m + 36 2. a2 + 4a +4

3. j2 + 4j +3 4. k2 + 7k +12

5. b2 + 8b +15 6. f2 + 6f +8

7. g2 + 9g +14 8. s2 + 10s +16

9. h2 + 5h +6 10. r2 + 7r +10

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- 2 )(y

- 2

8 = 1 x 8

8 = 2 x 4

-6 = -2 - 4

yy - 41st Step:

2nd Step:

Example 3: y2 - 6y + 8

-2y

-4y

(y - 4)

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- 3 )(y

- 3

21 = 1 x 21

21 = 3 x 7

-10 = -3 - 7

yy - 71st Step:

2nd Step:

Example 4: y2 - 10y + 21

-3y

-7y

(y - 7)

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Exercise


1. y2 – 3y + 2 6. y2 – 8y + 15

2. y2 – 5y + 6 7. y2 – 9y + 18

3. y2 – 7y + 12 8. y2 – 8y + 12

4. y2 – 7y + 10 9. y2 – 12y + 35

5. y2 – 9y + 14 10. y2 – 14y + 45

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Exercise:• Page 134• Ex.6.2B• 1 - 15

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FACTORISE ALGEBRAIC

EXPRESSIONS

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+9)(y

+9

Factorise a2 + 2ab - b2

36 = 1 x 3636 = 2 x 1836 = 3 x 1236 = 4 x 936 = 6 x 6

5 = 9 - 4

yy -41st Step:

2nd Step:

Example 5: y2 + 5y - 36

+9y

-4y

(y - 4)

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+7)(y

+7

21 = 1 x 2121 = 3 x 7

4 = 7 - 3

yy -31st Step:

2nd Step:

Example 6: y2 + 4y - 21

+7y

-3y

(y - 3)

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Exercise


1. y2 + y – 6 6. y2 + 4y – 12

2. y2 + 2y – 15 7. y2 + y – 42

3. y2 + 6y – 27 8. y2 + 4y – 21

4. y2 + 3y – 10 9. y2 + 7y – 30

5. y2 + 3y – 18 10. y2 + 10y – 24

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-5)(y

-5


10 = 1 x 1010 = 2 x 5

-3 = - 5 + 2

yy +21st Step:

2nd Step:

Example 7: y2 - 3y - 10

-5y

+2y

(y +2)

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-7)(y

-7


28 = 1 x 2828 = 2 x 1428 = 4 x 7

-3 = - 7 + 4

yy +41st Step:

2nd Step:

Example 8: y2 - 3y - 28

-7y

+4y

(y + 4)

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Exercise


1. y2 – 2y – 8 6. y2 – 5y – 36

2. y2 – 3y – 18 7. y2 – y – 12

3. y2 – 5y – 14 8. y2 – 2y – 24

4. y2 – 7y – 30 9. y2 – 6y – 7

5. y2 – y – 20 10. y2 – y – 6

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FACTORISE AND SIMPLIFY ALGEBRAIC FRACTIONS

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Simplify the following expressions:

Example 1:

x

xy

2

8

1

4

1

4y y4

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Example 2:

ab

ba

4

20 2

1

5

1

5a a5

a

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ExercisePage 138Ex. 6.2FNo. 1 abcd

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Simplify the following algebraic fractions to the lowest terms

Example 3:

6 4

4

g 6g + 42

3g + 2

2 ( 3g + 2 )

2(3 2)

4

g

3 2

2

g

Working Space

2

1

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Example 4:

c

ba

2

82 2a - 8b2

a - 4b

2 ( a – 4b )c

ba

2

)4(2

c

ba 4

Working Space

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ExercisePage 138Ex.6.2FNo. 2 only

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ADDITION OF ALGEBRAIC FRACTIONS

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Simplify the following:

Example 1:

7

2

7

3 xx

7

23 xx

7

5x

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Example 2:

3

42

3

3

gg

3

423

gg

3

73 g

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Exercise

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SUBTRACTION OF ALGEBRAIC FRACTIONS

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Example 1:

10

2

10

9 pp

10

29 pp

10

7 p

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Example 2:

hk

f

hk

f 5289

hk

f

89

hk

f 37

2f - 5

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Example 3:

abc

k

abc

k

2

27

2

78

abc

k

2

78

abc

k

2

9

7k + 2

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Exercise

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ExercisePage 139Ex. 6.3ANo. 1,2,3,5,10,11,12,

13,14,15

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ADD OR SUBTRACT TWO ALGEBRAIC FRACTIONS WITH

ONE DENOMINATOR AS A MULTIPLE OF THE OTHER

DENOMINATOR

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Example 1:

4

2

2

3 xx

4

2x

4

8x

4

x2

x2

x6

x2

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Example 2:

15

2

5

4 hh

15

212 hh

15

2h

15

x 3

x 3

h12

15

14h

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Example 3:

12

2

6

5 hh

12

210 hh

12

2h

12

x 2

x 2

h10

12

8h

3

2h

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Exercise

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ADD AND SUBTRACT ALGEBRAIC FRACTIONS WITHOUT ANY COMMON

FACTORS

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Example 1:

5

2

2

3 rp

10

415 rp

x 5 x 2x 5 x 2

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Example 2:n

f

m

58

n8

mn

mfmn 58

fm – 5m

x n x m

x n x m

mn

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Example 3:y

x

x

x 32

xy

xxyy 232

+ 3x2

x y x x

x y x x

xy

2y - xy

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Exercise

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ExercisePage 142&143ALLEx.6.3C

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MULTIPLICATION ALGEBRAIC FRACTIONS

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Example 1:

7

2

5

3

75

23

35

6

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Example 2:

q

p

n

m

qn

pm

nq

mp

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Example 3:

pr5

2

2

5pr

2

5pr

1

5p

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Example 4:)(

4)(

fe

g

h

yx

)(

4)(

feh

gyx

)(

)(4

feh

yxg

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Exercise

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ExercisePage 85Ex.7No. 1 to 12

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DIVISION OF TWO ALGEBRAIC FRACTIONS

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Example 1:

7

2

5

3

5

3

25

73

2

7

10

21

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Example 2:

pr5

2

1

5p

2

r

p5

1

p

r

52

1

p

r

10

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Example 3:

ed

c

4

5

3

2

d

c

3

2

53

42

d

ec

5

4e

d

ce

15

8

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ExercisePage 85Ex.7No. 13 to 29

NOTE MATH FORM 3 - ALGEBRAIC

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