1. Friday, January 25, 2013
AIRIL AHMAD

2. Friday, January 25, 2013
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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3. Friday, January 25, 2013
Example 2: Expand 3 (a - c)
3 ( a - c )
= 3 x a - 3 x c
= 3a - 3c
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4. Friday, January 25, 2013
Example 3: Expand 7g (m + 2n)
7g ( m + 2n )
= 7g x m + 7g x 2n
= 7gm + 14gn
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5. Friday, January 25, 2013
Example 4: Expand 4e ( r + 2s – 8t)
4e ( r + 2s - 8t )
= 4e x r + 4e x 2s - 4e x 8t
= 4er + 8es - 32et
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6. Friday, January 25, 2013
AIRIL AHMAD

7. Friday, January 25, 2013
ALGEBRAIC
EXPRESSIONS
EXPAND DOUBLE PAIRS OF BRACKETS
HJ. AIRIL AHMAD
AIRIL AHMAD

8. Friday, January 25, 2013
Expand double pairs of brackets
Example 1: Expand (w + v)(x + y)
( w + v ) ( x + y )
= wxx + w x y +vxx + vxy
= wx + wy + vx + vy
AIRIL AHMAD

9. Friday, January 25, 2013
Expand double pairs of brackets
Example 2: Expand (2e + f)(c + d)
( 2e + f ) ( c + d )
= 2e x c + 2e x d +fxc + fxd
= 2ce + 2de + cf + df
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10. Friday, January 25, 2013
Expand double pairs of brackets
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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11. Friday, January 25, 2013
EXPAND DOUBLE PAIRS OF BRACKETS
HJ. AIRIL AHMAD
AIRIL AHMAD

12. Friday, January 25, 2013
Expand double pairs of brackets
Example 1: Expand (r + 4)(r + 3)
( r + 4 ) ( r + 3 )
= rxr + r x 3 +4xr + 4x3
= r2 + 3r + 4r + 12
= r2 + 7r + 12
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13. Friday, January 25, 2013
Expand double pairs of brackets
Example 2: Expand (2e + f)(e + 2f)
( 2e + f ) ( e + 2f )
= 2e x e + 2e x 2f +fxe + f x 2f
= 2e2 + 4ef + ef + 2f2
= 2e2 + 5ef + 2f2
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14. Friday, January 25, 2013
EXERCISE
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15. Friday, January 25, 2013
Expand double pairs of brackets
Example 3: Expand (w + 4)(w - 3)
( w + 4 ) ( w - 3 )
= wxw + w x (-3) + 4 x w + 4 x (-3)
= w2 - 3w + 4w - 12
= w2 + w - 12
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16. Friday, January 25, 2013
EXPAND DOUBLE PAIRS OF BRACKETS
In general , (a+b)2= a2 + 2ab + b2 or
(a-b)2= a2 - 2ab + b2
HJ. AIRIL AHMAD
AIRIL AHMAD

17. Friday, January 25, 2013
Expand same pairs of brackets
Example 1: Expand (r + 4)2
( r + 4 ) ( r + 4 )
= rxr + r x 4 +4xr + 4x4
= r2 + 4r + 4r + 16
= r2 + 8r + 16 AIRIL AHMAD

18. Friday, January 25, 2013
Expand same pairs of brackets
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 AIRIL AHMAD

19. Friday, January 25, 2013
Expand same pairs of brackets
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 AIRIL AHMAD

20. Friday, January 25, 2013
State the factors of an algebraic term
Example 1: 2m
1, 2, m, 2m
Example 2: 4g
1, 2, 4, g, 2g, 4g
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21. Friday, January 25, 2013
State the factors of an algebraic term
Example 3: 12y
1, 2, 3, 4, 6,12, y, 2y, 3y, 4y, 6y, 12y
Example 4: 15k
1, 3, 5, 15, k, 3k, 5k, 15k
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22. Friday, January 25, 2013
Exercise
Write all the factors of the
following algebraic terms.
5j1. 5n 3. 2
2. 14 p 4. abc
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23. Friday, January 25, 2013
CONCEPT OF
FACTORISATION OF
ALGEBRAIC EXPRESSIONS
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AIRIL AHMAD

24. Friday, January 25, 2013
Highest Common Factor (HCF)
Find the HCF of the following algebraic terms.
Example 1 : Example 2 :
6h & 4gh 2pq & 4q
2 6h , 4gh 2 2pq , 4q
h 3h , 2gh q pq , 2q
3 , 2g p , 2
HCF = 2 x h HCF = 2 x q
= 2h = 2q
AIRIL AHMAD

25. Friday, January 25, 2013
Highest Common Factor (HCF)
Find the HCF of the following algebraic terms.
Example 3 : Example 4 :
14v2 & 8v 5cd2 & 10cd
2 14v2 , 8v 5 5cd2 , 10cd
v 7v2 , 4v cd cd2 , 2cd
7v , 4 d , 2
HCF = 2 x v HCF = 5 x cd
= 2v = 5cd
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26. Friday, January 25, 2013
EXERCISE
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27. Friday, January 25, 2013
AIRIL AHMAD

28. Friday, January 25, 2013
FACTORISE
ALGEBRAIC
EXPRESSIONS
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AIRIL AHMAD

29. Friday, January 25, 2013
Factorise the following algebraic term
Example 1: 2pq + 4qr
2 2pq + 4qr
q pq + 2qr
(p +
p 2r
2r)
AIRIL AHMAD

30. Friday, January 25, 2013
Factorise the following algebraic term
Example 2: mn – mn2
m mn - mn2
n n - n2
(1 - n
1 n)
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31. Friday, January 25, 2013
Factorise the following algebraic term
Example 3: 3ab + 6bc - 9b2
3 3ab + 6bc - 9b2
b ab + 2bc - 3b2
( a + 2c - 3b3b)
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32. Friday, January 25, 2013
EXERCISE
AIRIL AHMAD

33. Friday, January 25, 2013
FACTORISE
ALGEBRAIC
EXPRESSIONS
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AIRIL AHMAD

34. Friday, January 25, 2013
Factorise ab + ac +bd + cd
Example 9: ag + 3a + 2g + 6
(ag + 3) )+ (2g + 6 )
a
(g 3a a 2
(g 3)2
(a (g + 3) + 2 (g + 3)
)
AIRIL AHMAD

35. Friday, January 25, 2013
Factorise ab + ac +bd + cd
Example 10: mn + 4n + 2m + 8
( mn + 4n) +( 2(m +2 )
(m + 4) n
n 2m 8 4)
(n (m + 4) + 2 (m + 4)
)
AIRIL AHMAD

36. Friday, January 25, 2013
EXERCISE
AIRIL AHMAD

37. Friday, January 25, 2013
Exercise
Factorise the following algebraic expressions:
1. ac + ad + bc + bd
2. ax + ay + bx + by
3. gh + 2h + 3g + 6
4. a2 + ab + 2a + 2b
5. de + d2 + ef + df
AIRIL AHMAD

38. Friday, January 25, 2013
Factorise a2- b2
Example 11: a 2 – b2
( a 2 – - b2 ) )
(a + b
a
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39. Friday, January 25, 2013
Factorise a2- b2
Example 12: m 2 – n2
((m2 –- n2 ) )
m + n n
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40. Friday, January 25, 2013
Factorise a2- b2
Example 13: 16m2 – 25n2
16m2 – 25n2
( 42m2 – -5n ) )
(4m + 55n
4m n
2 2
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41. Friday, January 25, 2013
EXERCISE
AIRIL AHMAD

42. Friday, January 25, 2013
FACTORISE
ALGEBRAIC
EXPRESSIONS
AIRIL AHMAD
AIRIL AHMAD

43. Friday, January 25, 2013
Factorise a2 + 2ab + b2
Example 1: y2 + 8y + 16
4
1st Step:
16 = 1 x 16 (y 4)
+
+4y
16 = 2 x 8
16 = 4 x 4
2nd Step:
+
+4y
8= 4+ 4 (y + 4)
AIRIL AHMAD

44. Friday, January 25, 2013
Example 2: 7
2
y2 + 9y + 14
1st Step:
14 = 1 x 14 (y 2)
+
+2y
14 = 2 x 7
2nd Step:
+
+7y
9= 2+ 7
(y + 7)
AIRIL AHMAD

45. Friday, January 25, 2013
Exercise:
• Page 135
• Ex.6.2C
•1 - 8
AIRIL AHMAD

46. Friday, January 25, 2013
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
AIRIL AHMAD

47. Friday, January 25, 2013
Example 3: y2 - 6y + - 2
84
1st Step:
8= 1x8 (y -2)
-2y
8= 2x4
2nd Step: -4y
-6 = -2 - 4
(y - 4)
AIRIL AHMAD

48. Friday, January 25, 2013
Example 4: y2 - 10y + - 3
7
21
1st Step:
21 = 1 x 21 (y -3)
-3y
21 = 3 x 7
2nd Step: -7y
-10 = -3 - 7
(y - 7)
AIRIL AHMAD

49. Friday, January 25, 2013
Exercise
Factorise the following algebraic expressions:
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
AIRIL AHMAD

50. Friday, January 25, 2013
Exercise:
• Page 134
• Ex.6.2B
• 1 - 15
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51. Friday, January 25, 2013
FACTORISE
ALGEBRAIC
EXPRESSIONS
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AIRIL AHMAD

52. Friday, January 25, 2013
Factorise a2 + 2ab - b2
Example 5: y2 + 5y - 36
+9
-4
1st Step:
36 = 1 x 36
(y +9)
36 = 2 x 18 +9y
36 = 3 x 12
36 = 4x 9
36 = 6x 6
-4y
2 Step:
nd
5= 9- 4 (y - 4)
AIRIL AHMAD

53. Friday, January 25, 2013
Example 6: y2 + 4y - 21
+7
-3
1st Step:
21 = 1 x 21
(y +7)
21 = 3 x 7 +7y
2nd Step:
4= 7- 3
-3y
(y - 3)
AIRIL AHMAD

54. Friday, January 25, 2013
Exercise
Factorise the following algebraic expressions:
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
AIRIL AHMAD

55. Friday, January 25, 2013
Factorise a2 + 2ab + b2
Example 7: y2 - 3y - 10
+2
-5
1st Step:
10 = 1 x 10 (y -5)
10 = 2 x 5 -5y
2nd Step:
-3 = - 5 + 2 +2y
(y +2)
AIRIL AHMAD

56. Friday, January 25, 2013
Factorise a2 + 2ab + b2
Example 8: y2 - 3y - 28
+4
-7
1st Step:
28 = 1 x 28 (y -7)
28 = 2 x 14 -7y
28 = 4 x 7
2nd Step:
-3 = - 7 + 4 +4y
(y + 4)
AIRIL AHMAD

57. Friday, January 25, 2013
Exercise
Factorise the following algebraic expressions:
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
AIRIL AHMAD

58. Friday, January 25, 2013
FACTORISE AND SIMPLIFY
ALGEBRAIC FRACTIONS
AIRIL AHMAD
AIRIL AHMAD

59. Friday, January 25, 2013
Simplify the following expressions:
Example 1:
4
8 xy 4y
= = 4y
1
2x 1
AIRIL AHMAD

60. Friday, January 25, 2013
Example 2:
5 a 2
20a b 5a
4ab
=
1
= 5a
1
AIRIL AHMAD

61. Friday, January 25, 2013
Exercise
Page 138
Ex. 6.2F
No. 1 abcd
AIRIL AHMAD

62. Friday, January 25, 2013
Simplify the following algebraic
fractions to the lowest terms Working Space
Example 3:
1 2 6g + 4
6g + 4 2(3g + 2)
= 3g + 2
4 42
3g + 2 2 ( 3g + 2 )
=
2
AIRIL AHMAD

63. Friday, January 25, 2013
Working Space
Example 4:
2 2a - 8b
2a − 8b 2(a − 4b)
= a - 4b
2c 2c
2 ( a – 4b )
a − 4b
=
c AIRIL AHMAD

64. Friday, January 25, 2013
Exercise
Page 138
Ex.6.2F
No. 2 only
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65. Friday, January 25, 2013
ADDITION OF
ALGEBRAIC FRACTIONS
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AIRIL AHMAD

66. Friday, January 25, 2013
Simplify the following:
Example 1:
3x 2 x 3x + 2 x 5x
+ = =
7 7 7 7
AIRIL AHMAD

67. Friday, January 25, 2013
Example 2: g + 3 2g + 4
+
3 3
g + 3 + 2g + 4
=
3
3g + 7
=
3
AIRIL AHMAD

68. Friday, January 25, 2013
Exercise
AIRIL AHMAD

69. Friday, January 25, 2013
SUBTRACTION OF
ALGEBRAIC FRACTIONS
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AIRIL AHMAD

70. Friday, January 25, 2013
Simplify the following:
Example 1:
9p 2p 9p −2p 7p
− = =
10 10 10 10
AIRIL AHMAD

71. Friday, January 25, 2013
Example 2: 9 f +8 2 f +5
−
hk hk
9 f + 8 − 2f - 5
=
hk
7f +3
=
hk AIRIL AHMAD

72. Friday, January 25, 2013
8k + 7 7k − 2
Example 3: −
2abc 2abc
8k + 7 − 7k + 2
=
2abc
k+9
=
2abc AIRIL AHMAD

73. Friday, January 25, 2013
Exercise
AIRIL AHMAD

74. Friday, January 25, 2013
Exercise
Page 139
Ex. 6.3A
No. 1,2,3,5,10,11,12,
13,14,15
AIRIL AHMAD

75. Friday, January 25, 2013
ADD OR SUBTRACT TWO
ALGEBRAIC FRACTIONS WITH
ONE DENOMINATOR AS A
MULTIPLE OF THE OTHER
DENOMINATOR
AIRIL AHMAD
AIRIL AHMAD

76. Friday, January 25, 2013
Simplify the following:
Example 1: 3 xx2 2 x
+
2 x2 4
=6 x+ 2x
4 4
8x
= = 2x
4 AIRIL AHMAD

77. Friday, January 25, 2013
4h x 3 2h
Example 2: +
5 x 3 15
12h +2h
=
15 15
12h + 2h 14h
= =
15 15
AIRIL AHMAD

78. Friday, January 25, 2013
5hx 2 2 h
Example 3: −
6 x 2 12
10h −2h
=
12 12
10h − 2h 8h 2h
= = =
12 12 3
AIRIL AHMAD

79. Friday, January 25, 2013
Exercise
AIRIL AHMAD

80. Friday, January 25, 2013
ADD AND SUBTRACT
ALGEBRAIC FRACTIONS
WITHOUT ANY COMMON
FACTORS
AIRIL AHMAD
AIRIL AHMAD

81. Friday, January 25, 2013
Simplify the following:
Example 1:
3 p 5 2r 2
x x 15 p − 4r
− =
2x5 5x 2 10
AIRIL AHMAD

82. Friday, January 25, 2013
8x n f + x m
5
Example 2: −
m n
x nx m
8n − fm – 5m
=
mn
8n − fm − 5m
=
mn
AIRIL AHMAD

83. Friday, January 25, 2013
2 − x y 3 xx x
x
Example 3: +
xxy yx x
2y - xy + 3x2
=
xy
2 y − xy + 3x 2
=
xy
AIRIL AHMAD

84. Friday, January 25, 2013
Exercise
AIRIL AHMAD

85. Friday, January 25, 2013
Exercise
Page
142&143
ALL
Ex.6.3C
AIRIL AHMAD

86. Friday, January 25, 2013
MULTIPLICATION
ALGEBRAIC FRACTIONS
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AIRIL AHMAD

87. Friday, January 25, 2013
Simplify the following:
Example 1:
3 2 3× 2 6
× = =
5 7 5 × 7 35
AIRIL AHMAD

88. Friday, January 25, 2013
Example 2:
m p m × p mp
× = =
n q n × q nq
AIRIL AHMAD

89. Friday, January 25, 2013
5p
Example 3:
1
r
×5p = r ×5p 5 pr
2 =
2 2
AIRIL AHMAD

90. Friday, January 25, 2013
( x + y) 4g
Example 4: ×
h (e − f )
( x + y) × 4 g
=
h × (e − f )
4 g ( x + y)
=
h (e − f )
AIRIL AHMAD

91. Friday, January 25, 2013
Exercise
AIRIL AHMAD

92. Friday, January 25, 2013
Exercise
Page 85
Ex.7
No. 1 to 12
AIRIL AHMAD

93. Friday, January 25, 2013
DIVISION OF TWO
ALGEBRAIC FRACTIONS
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AIRIL AHMAD

94. Friday, January 25, 2013
Simplify the following:
Example 1:
×
3 2 3
÷ =
5 7 5
7 3 × 7 21
= =
2 5 × 2 10
AIRIL AHMAD

95. Friday, January 25, 2013
5p
Example 2: 1
r
2
×
÷5p =
r
2
1
5p
r ×1 r
= =
2×5 p 10 p
AIRIL AHMAD

96. Friday, January 25, 2013
Example 3:
÷ = ×
2c 5 2c
3d 4e 3d
4e
5
2c × 4e 8ce
= =
3d × 5 15d
AIRIL AHMAD

97. Friday, January 25, 2013
Exercise
Page 85
Ex.7
No. 13 to 29
AIRIL AHMAD