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Simplify each of the following expressions:

1. Simplify each of the following expressions:. ( i ). 3 x + 5 x – 2 x. = 6 x. (ii). – 4 y + 8 y – y. = 3 y. 1. Simplify each of the following expressions:. (iii). 12 + 4 n + 5 – 2 n – 8. 4 n – 2 n + 12 + 5 – 8 (group like terms). = 2 n + 9. (iv).

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Simplify each of the following expressions:

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  1. 1. Simplify each of the following expressions: (i) 3x + 5x – 2x = 6x (ii) – 4y + 8y – y = 3y

  2. 1. Simplify each of the following expressions: (iii) 12 + 4n + 5 – 2n – 8 4n – 2n + 12 + 5 – 8 (group like terms) = 2n + 9 (iv) 8a – 2b + 4a + 3b – 2a 8a + 4a – 2a – 2b + 3b (group like terms) = 10a + b

  3. 1. Simplify each of the following expressions: (v) 10x + 2 + 6x – 3 + 5x 10x + 6x + 5x – 3 + 2 (group like terms) = 21x – 1 (vi) 8c – b + 4a – a + 5b – 3c 4a – a + 5b – b + 8c – 3c (group like terms) = 3a + 4b + 5c

  4. 2. Simplify each of the following expressions: (i) 9p2 + 3p – 2p2 – 7p 9p2 + 3p – 2p2 – 7p = 7p2 – 4p (ii) 13y – 3y2 + 5y + 4y2 – 9y 13y – 3y2 + 5y + 4y2 – 9y = y2 + 9y

  5. 2. Simplify each of the following expressions: (iii) 5x2 + 3 – 2x + 3x2 + 7x – 5 5x2 + 3 – 2x + 3x2 + 7x – 5 = 8x2 + 5x – 2 (iv) – 2a + 4a2 + 6 – a2 + 8a – 5 – 2a + 4a2 + 6 – a2 + 8a – 5 = 3a2 + 6a + 1

  6. 2. Simplify each of the following expressions: (v) 12 + 8p – 4p2 – 3p – 5 + p2 12 + 8p – 4p2 – 3p – 5 + p2 = – 3p2 + 5p + 7 (vi) 7x2 – 9x + 13 + 5x – 2x2 – 6 7x2 – 9x + 13 + 5x – 2x2 – 6 = 5x2 – 4x + 7

  7. 3. Expand the following brackets and simplify: (i) (2x) (4x) + 3(4y) – 5y (2x) (4x) + 3(4y) – 5y = 8x2 + 12y – 5y = 8x2 + 7y (ii) 4(3p + 1) – 2(p – 2) 4(3p + 1) – 2(p – 2) = 12p + 4 – 2p + 4 = 10p + 8

  8. 3. Expand the following brackets and simplify: (iii) 7(k – 1) – 3(4k – 1) 7(k – 1) – 3(4k – 1) = 7k – 7 – 12k + 3 = – 5k – 4 (iv) 3x(x – 2) + 5(2x + 3) 3x(x – 2) + 5(2x + 3) = 3x2 – 6x + 10x + 15 = 3x2 + 4x + 15

  9. 3. Expand the following brackets and simplify: (v) 3(2a – 2) + 4a (3a – 1) + 7 3(2a – 2) + 4a (3a – 1) + 7 = 6a – 6 + 12a2 – 4a + 7 = 12a2 + 2a + 1 (vi) 6a(2x + 3) + 7 – (4a – 4) 6a(2x + 3) + 7 – (4a – 4) = 12ax + 18a + 7 – 4a + 4 = 12ax + 14a + 11

  10. 3. Expand the following brackets and simplify: (vii) 7x(3x – 2) – 14x + 3(3x + 2) 7x(3x – 2) – 14x + 3(3x + 2) = 21x2 – 14x – 14x + 9x + 6 = 21x2 – 19x + 6 (viii) – 4p(3p + 1) + 6 – (7 – 8p) – 4p(3p + 1) + 6 – (7 – 8p) = – 12p2 – 4p + 6 – 7 + 8p = – 12p2 + 4p – 1

  11. 4. Expand the following brackets and simplify: (i) (x + 1) (3x – 2) (x + 1) (3x – 2) = x(3x – 2) + 1(3x – 2) = 3x2 – 2x + 3x – 2 = 3x2 + x – 2

  12. 4. Expand the following brackets and simplify: (ii) (p + 5)(4p – 3) (p + 5)(4p – 3) = p(4p – 3) + 5(4p – 3) = 4p2 – 3p + 20p – 15 = 4p2 + 17p – 15

  13. 4. Expand the following brackets and simplify: (iii) (2x – 4) (3x + 2) (2x – 4) (3x + 2) = 2x(3x + 2) – 4(3x + 2) = 6x2 + 4x – 12x – 8 = 6x2 – 8x – 8

  14. 4. Expand the following brackets and simplify: (iv) (3k – 4) (5k – 1) (3k – 4) (5k – 1) = 3k(5k – 1) – 4(5k – 1) = 15k2 – 3k – 20k + 4 = 15k2 – 23k + 4

  15. 4. Expand the following brackets and simplify: (v) (5x – 2)(x + 1) – (x2 + 3x + 4) (5x – 2)(x + 1) – (x2 + 3x + 4) = 5x(x + 1) – 2(x + 1) – (x2 + 3x + 4) = 5x2 + 5x – 2x – 2 – x2 – 3x – 4 = 4x2 – 6

  16. 4. Expand the following brackets and simplify: (vi) (2p2 – 4p + 6) + (3 – p)(p + 4) (2p2 – 4p + 6) + (3 – p)(p + 4) = 2p2 – 4p + 6 + 3(p + 4) – p(p + 4) = 2p2 – 4p + 6 + 3p + 12 – p2 – 4p = p2 – 5p + 18

  17. 5. Expand the following brackets and simplify: (i) (x – 5)2 (x – 5)2 = (x – 5)(x – 5) = x(x – 5) – 5(x – 5) = x2 – 5x – 5x + 25 = x2 – 10x + 25

  18. 5. Expand the following brackets and simplify: (ii) (3p + 2)2 (3p + 2)2 = (3p + 2)(3p + 2) = 3p(3p + 2) + 2(3p + 2) = 9p2 + 6p + 6p + 4 = 9p2 + 12p + 4

  19. 5. Expand the following brackets and simplify: (iii) (4a– 2)2 (4a– 2)2 = (4a – 2)(4a – 2) = 4a(4a – 2) – 2(4a – 2) = 16a2 – 8a – 8a + 4 = 16a2 – 16a + 4

  20. 5. Expand the following brackets and simplify: (iv) (3 – 7k)2 (3 – 7k)2 = (3 – 7k)(3 – 7k) = 3(3 – 7k) – 7k(3 – 7k) = 9 – 21k – 21k + 49k2 = 49k2 – 42k + 9

  21. 6. Simplify the following: (i) (x + 3)(x + 4) (x + 3)(x + 4) = x(x + 4) + 3(x + 4) = x2 + 4x + 3x + 12 = x2 + 7x + 12

  22. 6. Simplify the following: (ii) 3a – 4 + 5 + 2a + 4 3a – 4 + 5 + 2a + 4 = 5a + 5

  23. 6. Simplify the following: (iii) 4(2k + 5) – k(2k + 6) – 3 4(2k + 5) – k(2k + 6) – 3 = 8k + 20 – 2k2 – 6k – 3 = – 2k2 + 2k + 17

  24. 6. Simplify the following: (iv) (a + 5) (a – 1) (a + 5) (a – 1) = a(a – 1) + 5(a – 1) = a2 – a + 5a – 5 = a2 + 4a – 5

  25. 6. Simplify the following: (v) 3(2a – 2) – 5 + 4a(3a – 1) 3(2a – 2) – 5 + 4a(3a – 1) = 6a – 6 – 5 + 12a2 – 4a = 12a2 + 2a – 11

  26. 6. Simplify the following: (vi) 5(x + 3) + 2(4x – 1) 5(x + 3) + 2(4x – 1) = 5x + 15 + 8x – 2 = 13x + 13

  27. 6. Simplify the following: (vii) – 6p + 2q – p + 4r – 4q + 2r – 6p + 2q – p + 4r – 4q + 2r = – 7p – 2q + 6r

  28. 6. Simplify the following: (viii) (x – 3)(x + 3) (x – 3)(x + 3) = x(x + 3) – 3(x + 3) = x2 + 3x – 3x – 9 = x2 – 9

  29. 6. Simplify the following: (ix) (y – 2)2 (y – 2)2 = (y – 2)(y – 2) = y(y – 2) – 2(y – 2) = y2 – 2y – 2y + 4 = y2 – 4y + 4

  30. 6. Simplify the following: (x) 3x2 + 2x – (x + 3)2 3x2 + 2x – (x + 3)2 = 3x2 + 2x – ((x + 3)(x + 3)) = 3x2 + 2x – (x(x + 3) + 3(x + 3)) = 3x2 + 2x – (x2 + 3x + 3x + 9) = 3x2 + 2x – x2 – 3x – 3x – 9 = 2x2 – 4x – 9

  31. 7. Find the sum of (2x – 1) and (x + 5) (2x – 1) + (x + 5) = 2x – 1 + x + 5 = 3x + 4

  32. 8. Find the product of (p + 7) and (3p − 2) (p + 7)(3p – 2) = p(3p – 2) + 7(3p – 2) = 3p2 – 2p + 21p – 14 = 3p2 + 19p – 14

  33. 9. If the length of a rectangle is (4x + 5) and its width is 7. (i) find the area of the rectangle, in terms of x. Area = length × width = (4x + 5) × 7 = 28x + 35

  34. 9. If the length of a rectangle is (4x + 5) and its width is 7. (ii) find the perimeter of the rectangle, in terms of x. Perimeter = 2(length) + 2(width) = 2(4x + 5) + 2(7) = 8x + 10 + 14 = 8x + 24

  35. 10. If the length of a rectangle is (2a − 1) and its width is (5 – a). (i) find the area of the rectangle, in terms of a. Area = length × width = (2a – 1)(5 – a) = 2a(5 – a) – 1(5 – a) = 10a – 2a2 – 5 + a = – 2a2 + 11a – 5

  36. 10. If the length of a rectangle is (2a − 1) and its width is (5 – a). (ii) find the perimeter of the rectangle, in terms of a. Perimeter = 2(length) + 2(width) = 2(2a – 1) + 2(5 – a) = 4a – 2 + 10 – 2a = 2a + 8

  37. 11. If the base of a triangle is 8 and its perpendicular height is (6x − 1), find the area of the triangle, in terms of x. Area = (base)( height) = (8) (6x – 1) = 4(6x – 1) = 24x – 4

  38. 12. Find the area of a circle of radius (3k – 2). Give your answer in terms ofkandπ. Area = πr2 = π × (3k – 2)2 = π × (3k – 2)(3k – 2) = π × (3k(3k – 2) – 2(3k – 2)) = π × (9k2 – 6k – 6k + 4) = π × (9k2 – 12k + 4) = (9k2 – 12k + 4)π

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