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NOTE MATH FORM 3 - ALGEBRAIC

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NOTE MATH FORM 3 - ALGEBRAIC

1. 1. Friday, January 25, 2013 AIRIL AHMAD
2. 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 AIRIL AHMAD
3. 3. Friday, January 25, 2013 Example 2: Expand 3 (a - c) 3 ( a - c ) = 3 x a - 3 x c = 3a - 3c AIRIL AHMAD
4. 4. Friday, January 25, 2013 Example 3: Expand 7g (m + 2n) 7g ( m + 2n ) = 7g x m + 7g x 2n = 7gm + 14gn AIRIL AHMAD
5. 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 AIRIL AHMAD
6. 6. Friday, January 25, 2013 AIRIL AHMAD
7. 7. Friday, January 25, 2013 ALGEBRAIC EXPRESSIONS EXPAND DOUBLE PAIRS OF BRACKETS HJ. AIRIL AHMAD AIRIL AHMAD
8. 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. 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 AIRIL AHMAD
10. 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 AIRIL AHMAD
11. 11. Friday, January 25, 2013 EXPAND DOUBLE PAIRS OF BRACKETS HJ. AIRIL AHMAD AIRIL AHMAD
12. 12. Friday, January 25, 2013Expand 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 AIRIL AHMAD
13. 13. Friday, January 25, 2013Expand 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 AIRIL AHMAD
14. 14. Friday, January 25, 2013 EXERCISE AIRIL AHMAD
15. 15. Friday, January 25, 2013Expand 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 AIRIL AHMAD
16. 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. 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. 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. 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. 20. Friday, January 25, 2013State the factors of an algebraic termExample 1: 2m 1, 2, m, 2mExample 2: 4g 1, 2, 4, g, 2g, 4g AIRIL AHMAD
21. 21. Friday, January 25, 2013State the factors of an algebraic termExample 3: 12y 1, 2, 3, 4, 6,12, y, 2y, 3y, 4y, 6y, 12yExample 4: 15k 1, 3, 5, 15, k, 3k, 5k, 15k AIRIL AHMAD
22. 22. Friday, January 25, 2013ExerciseWrite all the factors of the following algebraic terms. 5j1. 5n 3. 2 2. 14 p 4. abc AIRIL AHMAD
23. 23. Friday, January 25, 2013 CONCEPT OF FACTORISATION OFALGEBRAIC EXPRESSIONS AIRIL AHMAD AIRIL AHMAD
24. 24. Friday, January 25, 2013Highest 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. 25. Friday, January 25, 2013Highest 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 AIRIL AHMAD
26. 26. Friday, January 25, 2013 EXERCISE AIRIL AHMAD
27. 27. Friday, January 25, 2013 AIRIL AHMAD
28. 28. Friday, January 25, 2013 FACTORISE ALGEBRAIC EXPRESSIONS AIRIL AHMAD AIRIL AHMAD
29. 29. Friday, January 25, 2013Factorise the following algebraic termExample 1: 2pq + 4qr 2 2pq + 4qr q pq + 2qr (p + p 2r 2r) AIRIL AHMAD
30. 30. Friday, January 25, 2013Factorise the following algebraic termExample 2: mn – mn2 m mn - mn2 n n - n2 (1 - n 1 n) AIRIL AHMAD
31. 31. Friday, January 25, 2013Factorise the following algebraic termExample 3: 3ab + 6bc - 9b2 3 3ab + 6bc - 9b2 b ab + 2bc - 3b2 ( a + 2c - 3b3b) AIRIL AHMAD
32. 32. Friday, January 25, 2013 EXERCISE AIRIL AHMAD
33. 33. Friday, January 25, 2013 FACTORISE ALGEBRAIC EXPRESSIONS AIRIL AHMAD AIRIL AHMAD
34. 34. Friday, January 25, 2013Factorise ab + ac +bd + cdExample 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. 35. Friday, January 25, 2013Factorise ab + ac +bd + cdExample 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. 36. Friday, January 25, 2013 EXERCISE AIRIL AHMAD
37. 37. Friday, January 25, 2013Exercise 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. 38. Friday, January 25, 2013Factorise a2- b2Example 11: a 2 – b2 ( a 2 – - b2 ) ) (a + b a AIRIL AHMAD
39. 39. Friday, January 25, 2013Factorise a2- b2Example 12: m 2 – n2 ((m2 –- n2 ) ) m + n n AIRIL AHMAD
40. 40. Friday, January 25, 2013Factorise a2- b2Example 13: 16m2 – 25n2 16m2 – 25n2 ( 42m2 – -5n ) ) (4m + 55n 4m n 2 2 AIRIL AHMAD
41. 41. Friday, January 25, 2013 EXERCISE AIRIL AHMAD
42. 42. Friday, January 25, 2013 FACTORISE ALGEBRAIC EXPRESSIONS AIRIL AHMAD AIRIL AHMAD
43. 43. Friday, January 25, 2013Factorise a2 + 2ab + b2Example 1: y2 + 8y + 16 41st Step: 16 = 1 x 16 (y 4) + +4y 16 = 2 x 8 16 = 4 x 42nd Step: + +4y 8= 4+ 4 (y + 4) AIRIL AHMAD
44. 44. Friday, January 25, 2013Example 2: 7 2 y2 + 9y + 141st Step: 14 = 1 x 14 (y 2) + +2y 14 = 2 x 72nd Step: + +7y 9= 2+ 7 (y + 7) AIRIL AHMAD
45. 45. Friday, January 25, 2013 Exercise: • Page 135 • Ex.6.2C •1 - 8 AIRIL AHMAD
46. 46. Friday, January 25, 2013Factorise 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. 47. Friday, January 25, 2013Example 3: y2 - 6y + - 2 841st Step: 8= 1x8 (y -2) -2y 8= 2x42nd Step: -4y -6 = -2 - 4 (y - 4) AIRIL AHMAD
48. 48. Friday, January 25, 2013Example 4: y2 - 10y + - 3 7 211st Step: 21 = 1 x 21 (y -3) -3y 21 = 3 x 72nd Step: -7y -10 = -3 - 7 (y - 7) AIRIL AHMAD
49. 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. 50. Friday, January 25, 2013 Exercise: • Page 134 • Ex.6.2B • 1 - 15 AIRIL AHMAD
51. 51. Friday, January 25, 2013 FACTORISE ALGEBRAIC EXPRESSIONS AIRIL AHMAD AIRIL AHMAD
52. 52. Friday, January 25, 2013Factorise a2 + 2ab - b2Example 5: y2 + 5y - 36 +9 -41st Step: 36 = 1 x 36 (y +9) 36 = 2 x 18 +9y 36 = 3 x 12 36 = 4x 9 36 = 6x 6 -4y2 Step: nd 5= 9- 4 (y - 4) AIRIL AHMAD
53. 53. Friday, January 25, 2013Example 6: y2 + 4y - 21 +7 -31st Step: 21 = 1 x 21 (y +7) 21 = 3 x 7 +7y2nd Step: 4= 7- 3 -3y (y - 3) AIRIL AHMAD
54. 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. 55. Friday, January 25, 2013Factorise a2 + 2ab + b2Example 7: y2 - 3y - 10 +2 -51st Step: 10 = 1 x 10 (y -5) 10 = 2 x 5 -5y2nd Step: -3 = - 5 + 2 +2y (y +2) AIRIL AHMAD
56. 56. Friday, January 25, 2013Factorise a2 + 2ab + b2Example 8: y2 - 3y - 28 +4 -71st Step: 28 = 1 x 28 (y -7) 28 = 2 x 14 -7y 28 = 4 x 72nd Step: -3 = - 7 + 4 +4y (y + 4) AIRIL AHMAD
57. 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. 58. Friday, January 25, 2013FACTORISE AND SIMPLIFY ALGEBRAIC FRACTIONS AIRIL AHMAD AIRIL AHMAD
59. 59. Friday, January 25, 2013 Simplify the following expressions: Example 1: 4 8 xy 4y = = 4y 1 2x 1 AIRIL AHMAD
60. 60. Friday, January 25, 2013 Example 2: 5 a 2 20a b 5a 4ab = 1 = 5a 1 AIRIL AHMAD
61. 61. Friday, January 25, 2013 Exercise Page 138 Ex. 6.2F No. 1 abcd AIRIL AHMAD
62. 62. Friday, January 25, 2013Simplify the following algebraicfractions to the lowest terms Working Space Example 3: 1 2 6g + 46g + 4 2(3g + 2) = 3g + 2 4 42 3g + 2 2 ( 3g + 2 ) = 2 AIRIL AHMAD
63. 63. Friday, January 25, 2013 Working Space Example 4: 2 2a - 8b2a − 8b 2(a − 4b) = a - 4b 2c 2c 2 ( a – 4b ) a − 4b = c AIRIL AHMAD
64. 64. Friday, January 25, 2013 Exercise Page 138 Ex.6.2F No. 2 only AIRIL AHMAD
65. 65. Friday, January 25, 2013 ADDITION OF ALGEBRAIC FRACTIONS AIRIL AHMAD AIRIL AHMAD
66. 66. Friday, January 25, 2013Simplify the following:Example 1: 3x 2 x 3x + 2 x 5x + = = 7 7 7 7 AIRIL AHMAD
67. 67. Friday, January 25, 2013Example 2: g + 3 2g + 4 + 3 3 g + 3 + 2g + 4 = 3 3g + 7 = 3 AIRIL AHMAD
68. 68. Friday, January 25, 2013 Exercise AIRIL AHMAD
69. 69. Friday, January 25, 2013 SUBTRACTION OF ALGEBRAIC FRACTIONS AIRIL AHMAD AIRIL AHMAD
70. 70. Friday, January 25, 2013Simplify the following:Example 1: 9p 2p 9p −2p 7p − = = 10 10 10 10 AIRIL AHMAD
71. 71. Friday, January 25, 2013Example 2: 9 f +8 2 f +5 − hk hk 9 f + 8 − 2f - 5 = hk 7f +3 = hk AIRIL AHMAD
72. 72. Friday, January 25, 2013 8k + 7 7k − 2Example 3: − 2abc 2abc 8k + 7 − 7k + 2 = 2abc k+9 = 2abc AIRIL AHMAD
73. 73. Friday, January 25, 2013 Exercise AIRIL AHMAD
74. 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. 75. Friday, January 25, 2013 ADD OR SUBTRACT TWOALGEBRAIC FRACTIONS WITH ONE DENOMINATOR AS A MULTIPLE OF THE OTHER DENOMINATOR AIRIL AHMAD AIRIL AHMAD
76. 76. Friday, January 25, 2013Simplify the following:Example 1: 3 xx2 2 x + 2 x2 4 =6 x+ 2x 4 4 8x = = 2x 4 AIRIL AHMAD
77. 77. Friday, January 25, 2013 4h x 3 2hExample 2: + 5 x 3 15 12h +2h = 15 15 12h + 2h 14h = = 15 15 AIRIL AHMAD
78. 78. Friday, January 25, 2013 5hx 2 2 hExample 3: − 6 x 2 12 10h −2h = 12 12 10h − 2h 8h 2h = = = 12 12 3 AIRIL AHMAD
79. 79. Friday, January 25, 2013 Exercise AIRIL AHMAD
80. 80. Friday, January 25, 2013 ADD AND SUBTRACT ALGEBRAIC FRACTIONS WITHOUT ANY COMMON FACTORS AIRIL AHMAD AIRIL AHMAD
81. 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. 82. Friday, January 25, 2013 8x n f + x m 5Example 2: − m n x nx m 8n − fm – 5m = mn 8n − fm − 5m = mn AIRIL AHMAD
83. 83. Friday, January 25, 2013 2 − x y 3 xx x xExample 3: + xxy yx x 2y - xy + 3x2 = xy 2 y − xy + 3x 2 = xy AIRIL AHMAD
84. 84. Friday, January 25, 2013 Exercise AIRIL AHMAD
85. 85. Friday, January 25, 2013 Exercise Page 142&143 ALL Ex.6.3C AIRIL AHMAD
86. 86. Friday, January 25, 2013 MULTIPLICATION ALGEBRAIC FRACTIONS AIRIL AHMAD AIRIL AHMAD
87. 87. Friday, January 25, 2013 Simplify the following: Example 1: 3 2 3× 2 6 × = = 5 7 5 × 7 35 AIRIL AHMAD
88. 88. Friday, January 25, 2013 Example 2:m p m × p mp × = =n q n × q nq AIRIL AHMAD
89. 89. Friday, January 25, 2013 5p Example 3: 1r ×5p = r ×5p 5 pr2 = 2 2 AIRIL AHMAD
90. 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. 91. Friday, January 25, 2013 Exercise AIRIL AHMAD
92. 92. Friday, January 25, 2013 Exercise Page 85 Ex.7 No. 1 to 12 AIRIL AHMAD
93. 93. Friday, January 25, 2013 DIVISION OF TWO ALGEBRAIC FRACTIONS AIRIL AHMAD AIRIL AHMAD
94. 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. 95. Friday, January 25, 2013 5p Example 2: 1r2 × ÷5p = r 2 1 5p r ×1 r = = 2×5 p 10 p AIRIL AHMAD
96. 96. Friday, January 25, 2013 Example 3: ÷ = × 2c 5 2c 3d 4e 3d 4e 5 2c × 4e 8ce = = 3d × 5 15d AIRIL AHMAD
97. 97. Friday, January 25, 2013 Exercise Page 85 Ex.7 No. 13 to 29 AIRIL AHMAD