The document provides examples of expanding and factorizing algebraic expressions involving single and double pairs of brackets. It begins with examples of expanding single pairs of brackets by distributing terms. It then shows examples of expanding double pairs of brackets by distributing the first term over the second pair of brackets. The document also covers finding the highest common factor of algebraic terms and factorizing expressions involving addition or subtraction of like terms.

1. Friday, January 25, 2013
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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
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7. Friday, January 25, 2013
ALGEBRAIC
EXPRESSIONS
EXPAND DOUBLE PAIRS OF BRACKETS
HJ. AIRIL AHMAD
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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
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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
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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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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
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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
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28. Friday, January 25, 2013
FACTORISE
ALGEBRAIC
EXPRESSIONS
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29. Friday, January 25, 2013
Factorise the following algebraic term
Example 1: 2pq + 4qr
2 2pq + 4qr
q pq + 2qr
(p +
p 2r
2r)
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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
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33. Friday, January 25, 2013
FACTORISE
ALGEBRAIC
EXPRESSIONS
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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)
)
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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)
)
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36. Friday, January 25, 2013
EXERCISE
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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
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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
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42. Friday, January 25, 2013
FACTORISE
ALGEBRAIC
EXPRESSIONS
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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)
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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)
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45. Friday, January 25, 2013
Exercise:
• Page 135
• Ex.6.2C
•1 - 8
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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
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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)
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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)
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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
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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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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)
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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)
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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
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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)
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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)
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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
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58. Friday, January 25, 2013
FACTORISE AND SIMPLIFY
ALGEBRAIC FRACTIONS
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59. Friday, January 25, 2013
Simplify the following expressions:
Example 1:
4
8 xy 4y
= = 4y
1
2x 1
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60. Friday, January 25, 2013
Example 2:
5 a 2
20a b 5a
4ab
=
1
= 5a
1
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61. Friday, January 25, 2013
Exercise
Page 138
Ex. 6.2F
No. 1 abcd
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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
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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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66. Friday, January 25, 2013
Simplify the following:
Example 1:
3x 2 x 3x + 2 x 5x
+ = =
7 7 7 7
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67. Friday, January 25, 2013
Example 2: g + 3 2g + 4
+
3 3
g + 3 + 2g + 4
=
3
3g + 7
=
3
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68. Friday, January 25, 2013
Exercise
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69. Friday, January 25, 2013
SUBTRACTION OF
ALGEBRAIC FRACTIONS
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70. Friday, January 25, 2013
Simplify the following:
Example 1:
9p 2p 9p −2p 7p
− = =
10 10 10 10
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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
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74. Friday, January 25, 2013
Exercise
Page 139
Ex. 6.3A
No. 1,2,3,5,10,11,12,
13,14,15
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75. Friday, January 25, 2013
ADD OR SUBTRACT TWO
ALGEBRAIC FRACTIONS WITH
ONE DENOMINATOR AS A
MULTIPLE OF THE OTHER
DENOMINATOR
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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
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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
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79. Friday, January 25, 2013
Exercise
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80. Friday, January 25, 2013
ADD AND SUBTRACT
ALGEBRAIC FRACTIONS
WITHOUT ANY COMMON
FACTORS
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81. Friday, January 25, 2013
Simplify the following:
Example 1:
3 p 5 2r 2
x x 15 p − 4r
− =
2x5 5x 2 10
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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
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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
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84. Friday, January 25, 2013
Exercise
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85. Friday, January 25, 2013
Exercise
Page
142&143
ALL
Ex.6.3C
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86. Friday, January 25, 2013
MULTIPLICATION
ALGEBRAIC FRACTIONS
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87. Friday, January 25, 2013
Simplify the following:
Example 1:
3 2 3× 2 6
× = =
5 7 5 × 7 35
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88. Friday, January 25, 2013
Example 2:
m p m × p mp
× = =
n q n × q nq
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89. Friday, January 25, 2013
5p
Example 3:
1
r
×5p = r ×5p 5 pr
2 =
2 2
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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 )
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91. Friday, January 25, 2013
Exercise
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92. Friday, January 25, 2013
Exercise
Page 85
Ex.7
No. 1 to 12
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93. Friday, January 25, 2013
DIVISION OF TWO
ALGEBRAIC FRACTIONS
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94. Friday, January 25, 2013
Simplify the following:
Example 1:
×
3 2 3
÷ =
5 7 5
7 3 × 7 21
= =
2 5 × 2 10
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95. Friday, January 25, 2013
5p
Example 2: 1
r
2
×
÷5p =
r
2
1
5p
r ×1 r
= =
2×5 p 10 p
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96. Friday, January 25, 2013
Example 3:
÷ = ×
2c 5 2c
3d 4e 3d
4e
5
2c × 4e 8ce
= =
3d × 5 15d
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97. Friday, January 25, 2013
Exercise
Page 85
Ex.7
No. 13 to 29
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