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9.1 – Multiplying & Dividing Rational Expressions.

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1 9.1 – Multiplying & Dividing Rational Expressions

2 Ex. 1 Simplify. a. 30bc 12b 2

3 9.1 – Multiplying & Dividing Rational Expressions Ex. 1 Simplify. a. 30bc 12b 2 30bc = 12b 2

4 9.1 – Multiplying & Dividing Rational Expressions Ex. 1 Simplify. a. 30bc 12b 2 30bc = 6·5·b·c 12b 2

5 9.1 – Multiplying & Dividing Rational Expressions Ex. 1 Simplify. a. 30bc 12b 2 30bc = 6·5·b·c 12b 2 6·2·b·b

6 9.1 – Multiplying & Dividing Rational Expressions Ex. 1 Simplify. a. 30bc 12b 2 30bc = 6·5·b·c 12b 2 6·2·b·b

7 9.1 – Multiplying & Dividing Rational Expressions Ex. 1 Simplify. a. 30bc 12b 2 30bc = 6·5·b·c 12b 2 6·2·b·b

8 9.1 – Multiplying & Dividing Rational Expressions Ex. 1 Simplify. a. 30bc 12b 2 30bc = 6·5·b·c 12b 2 6·2·b·b

9 9.1 – Multiplying & Dividing Rational Expressions Ex. 1 Simplify. a. 30bc 12b 2 30bc = 6·5·b·c = 5c 12b 2 6·2·b·b 2b

10 9.1 – Multiplying & Dividing Rational Expressions Ex. 1 Simplify. a. 30bc 12b 2 30bc = 6·5·b·c = 5c 12b 2 6·2·b·b 2b b. 2x 2 – 10x x 2 – 4x – 5

11 9.1 – Multiplying & Dividing Rational Expressions Ex. 1 Simplify. a. 30bc 12b 2 30bc = 6·5·b·c = 5c 12b 2 6·2·b·b 2b b. 2x 2 – 10x x 2 – 4x – 5 2x 2 – 10x = x 2 – 4x – 5

12 9.1 – Multiplying & Dividing Rational Expressions Ex. 1 Simplify. a. 30bc 12b 2 30bc = 6·5·b·c = 5c 12b 2 6·2·b·b 2b b. 2x 2 – 10x x 2 – 4x – 5 2x 2 – 10x = x 2 – 4x – 5

13 9.1 – Multiplying & Dividing Rational Expressions Ex. 1 Simplify. a. 30bc 12b 2 30bc = 6·5·b·c = 5c 12b 2 6·2·b·b 2b b. 2x 2 – 10x x 2 – 4x – 5 2x 2 – 10x = 2x(x – 5) x 2 – 4x – 5

14 9.1 – Multiplying & Dividing Rational Expressions Ex. 1 Simplify. a. 30bc 12b 2 30bc = 6·5·b·c = 5c 12b 2 6·2·b·b 2b b. 2x 2 – 10x x 2 – 4x – 5 2x 2 – 10x = 2x(x – 5) x 2 – 4x – 5

15 9.1 – Multiplying & Dividing Rational Expressions Ex. 1 Simplify. a. 30bc 12b 2 30bc = 6·5·b·c = 5c 12b 2 6·2·b·b 2b b. 2x 2 – 10x x 2 – 4x – 5 2x 2 – 10x = 2x(x – 5) x 2 – 4x – 5 (x – 5)(x + 1)

16 9.1 – Multiplying & Dividing Rational Expressions Ex. 1 Simplify. a. 30bc 12b 2 30bc = 6·5·b·c = 5c 12b 2 6·2·b·b 2b b. 2x 2 – 10x x 2 – 4x – 5 2x 2 – 10x = 2x(x – 5) x 2 – 4x – 5 (x – 5)(x + 1)

17 9.1 – Multiplying & Dividing Rational Expressions Ex. 1 Simplify. a. 30bc 12b 2 30bc = 6·5·b·c = 5c 12b 2 6·2·b·b 2b b. 2x 2 – 10x x 2 – 4x – 5 2x 2 – 10x = 2x(x – 5) x 2 – 4x – 5 (x – 5)(x + 1)

18 9.1 – Multiplying & Dividing Rational Expressions Ex. 1 Simplify. a. 30bc 12b 2 30bc = 6·5·b·c = 5c 12b 2 6·2·b·b 2b b. 2x 2 – 10x x 2 – 4x – 5 2x 2 – 10x = 2x(x – 5) x 2 – 4x – 5 (x – 5)(x + 1)

19 9.1 – Multiplying & Dividing Rational Expressions Ex. 1 Simplify. a. 30bc 12b 2 30bc = 6·5·b·c = 5c 12b 2 6·2·b·b 2b b. 2x 2 – 10x x 2 – 4x – 5 2x 2 – 10x = 2x(x – 5) x 2 – 4x – 5 (x – 5)(x + 1) = 2x x + 1

20 Ex. 2 Simplify each expression. a. 4a. 15b 2 5b 16a 3

21 Ex. 2 Simplify each expression. a. 4a. 15b 2 5b 16a 3 4a. 15b 2 = 5b 16a 3

22 Ex. 2 Simplify each expression. a. 4a. 15b 2 5b 16a 3 4a. 15b 2 = 4a · 15b 2 5b 16a 3

23 Ex. 2 Simplify each expression. a. 4a. 15b 2 5b 16a 3 4a. 15b 2 = 4a · 15b 2 5b 16a 3 5b · 16a 3

24 Ex. 2 Simplify each expression. a. 4a. 15b 2 5b 16a 3 4a. 15b 2 = 4a · 15b 2 5b 16a 3 5b · 16a 3 = 2·2·3·5·a·b·b

25 Ex. 2 Simplify each expression. a. 4a. 15b 2 5b 16a 3 4a. 15b 2 = 4a · 15b 2 5b 16a 3 5b · 16a 3 = 2·2·3·5·a·b·b 2·2·2·2·5·a·a·a·b

26 Ex. 2 Simplify each expression. a. 4a. 15b 2 5b 16a 3 4a. 15b 2 = 4a · 15b 2 5b 16a 3 5b · 16a 3 = 2·2·3·5·a·b·b 2·2·2·2·5·a·a·a·b

27 Ex. 2 Simplify each expression. a. 4a. 15b 2 5b 16a 3 4a. 15b 2 = 4a · 15b 2 5b 16a 3 5b · 16a 3 = 2·2·3·5·a·b·b 2·2·2·2·5·a·a·a·b

28 Ex. 2 Simplify each expression. a. 4a. 15b 2 5b 16a 3 4a. 15b 2 = 4a · 15b 2 5b 16a 3 5b · 16a 3 = 2·2·3·5·a·b·b 2·2·2·2·5·a·a·a·b = 3·b 2·2·a·a

29 Ex. 2 Simplify each expression. a. 4a. 15b 2 5b 16a 3 4a. 15b 2 = 4a · 15b 2 5b 16a 3 5b · 16a 3 = 2·2·3·5·a·b·b 2·2·2·2·5·a·a·a·b = 3·b = 3b 2·2·a·a 4a 2

30 b. x 2 + 2x – 8. 3x + 3 x 2 + 4x + 3 x – 2

31 b. x 2 + 2x – 8. 3x + 3 x 2 + 4x + 3 x – 2 x 2 + 2x – 8. 3x + 3 = x 2 + 4x + 3 x – 2

32 b. x 2 + 2x – 8. 3x + 3 x 2 + 4x + 3 x – 2 x 2 + 2x – 8. 3x + 3 = (x + 4)(x – 2) x 2 + 4x + 3 x – 2

33 b. x 2 + 2x – 8. 3x + 3 x 2 + 4x + 3 x – 2 x 2 + 2x – 8. 3x + 3 = (x + 4)(x – 2) x 2 + 4x + 3 x – 2 (x + 3)(x + 1)

34 b. x 2 + 2x – 8. 3x + 3 x 2 + 4x + 3 x – 2 x 2 + 2x – 8. 3x + 3 = (x + 4)(x – 2). 3(x + 1) x 2 + 4x + 3 x – 2 (x + 3)(x + 1)

35 b. x 2 + 2x – 8. 3x + 3 x 2 + 4x + 3 x – 2 x 2 + 2x – 8. 3x + 3 = (x + 4)(x – 2). 3(x + 1) x 2 + 4x + 3 x – 2 (x + 3)(x + 1) (x – 2)

36 b. x 2 + 2x – 8. 3x + 3 x 2 + 4x + 3 x – 2 x 2 + 2x – 8. 3x + 3 = (x + 4)(x – 2). 3(x + 1) x 2 + 4x + 3 x – 2 (x + 3)(x + 1)(x – 2) = 3(x + 4)(x – 2)(x + 1)

37 b. x 2 + 2x – 8. 3x + 3 x 2 + 4x + 3 x – 2 x 2 + 2x – 8. 3x + 3 = (x + 4)(x – 2). 3(x + 1) x 2 + 4x + 3 x – 2 (x + 3)(x + 1)(x – 2) = 3(x + 4)(x – 2)(x + 1) (x + 3)(x + 1)(x – 2)

38 b. x 2 + 2x – 8. 3x + 3 x 2 + 4x + 3 x – 2 x 2 + 2x – 8. 3x + 3 = (x + 4)(x – 2). 3(x + 1) x 2 + 4x + 3 x – 2 (x + 3)(x + 1)(x – 2) = 3(x + 4)(x – 2)(x + 1) (x + 3)(x + 1)(x – 2)

39 b. x 2 + 2x – 8. 3x + 3 x 2 + 4x + 3 x – 2 x 2 + 2x – 8. 3x + 3 = (x + 4)(x – 2). 3(x + 1) x 2 + 4x + 3 x – 2 (x + 3)(x + 1)(x – 2) = 3(x + 4)(x – 2)(x + 1) (x + 3)(x + 1)(x – 2)

40 b. x 2 + 2x – 8. 3x + 3 x 2 + 4x + 3 x – 2 x 2 + 2x – 8. 3x + 3 = (x + 4)(x – 2). 3(x + 1) x 2 + 4x + 3 x – 2 (x + 3)(x + 1)(x – 2) = 3(x + 4)(x – 2)(x + 1) (x + 3)(x + 1)(x – 2)

41 b. x 2 + 2x – 8. 3x + 3 x 2 + 4x + 3 x – 2 x 2 + 2x – 8. 3x + 3 = (x + 4)(x – 2). 3(x + 1) x 2 + 4x + 3 x – 2 (x + 3)(x + 1)(x – 2) = 3(x + 4)(x – 2)(x + 1) (x + 3)(x + 1)(x – 2)

42 b. x 2 + 2x – 8. 3x + 3 x 2 + 4x + 3 x – 2 x 2 + 2x – 8. 3x + 3 = (x + 4)(x – 2). 3(x + 1) x 2 + 4x + 3 x – 2 (x + 3)(x + 1)(x – 2) = 3(x + 4)(x – 2)(x + 1) (x + 3)(x + 1)(x – 2) = 3(x + 4) (x + 3)

43 b. x 2 + 2x – 8. 3x + 3 x 2 + 4x + 3 x – 2 x 2 + 2x – 8. 3x + 3 = (x + 4)(x – 2). 3(x + 1) x 2 + 4x + 3 x – 2 (x + 3)(x + 1)(x – 2) = 3(x + 4)(x – 2)(x + 1) (x + 3)(x + 1)(x – 2) = 3(x + 4) = 3x + 12 (x + 3) x + 3

44 Ex. 3 Simplify each expression. a. 4x 2 y ÷ 2xy 2 15a 3 b 3 5ab 3

45 Ex. 3 Simplify each expression. a. 4x 2 y ÷ 2xy 2 15a 3 b 3 5ab 3 4x 2 y ÷ 2xy 2 = 15a 3 b 3 5ab 3

46 Ex. 3 Simplify each expression. a. 4x 2 y ÷ 2xy 2 15a 3 b 3 5ab 3 4x 2 y ÷ 2xy 2 = 4x 2 y 15a 3 b 3 5ab 3 15a 3 b 3

47 Ex. 3 Simplify each expression. a. 4x 2 y ÷ 2xy 2 15a 3 b 3 5ab 3 4x 2 y ÷ 2xy 2 = 4x 2 y. 5ab 3 15a 3 b 3 5ab 3 15a 3 b 3 2xy 2

48 Ex. 3 Simplify each expression. a. 4x 2 y ÷ 2xy 2 15a 3 b 3 5ab 3 4x 2 y ÷ 2xy 2 = 4x 2 y. 5ab 3 15a 3 b 3 5ab 3 15a 3 b 3 2xy 2

49 Ex. 3 Simplify each expression. a. 4x 2 y ÷ 2xy 2 15a 3 b 3 5ab 3 4x 2 y ÷ 2xy 2 = 4x 2 y. 5ab 3 15a 3 b 3 5ab 3 15a 3 b 3 2xy 2 = 2·2·5·a·b·b·b·x·x·y

50 Ex. 3 Simplify each expression. a. 4x 2 y ÷ 2xy 2 15a 3 b 3 5ab 3 4x 2 y ÷ 2xy 2 = 4x 2 y. 5ab 3 15a 3 b 3 5ab 3 15a 3 b 3 2xy 2 = 2·2·5·a·b·b·b·x·x·y 2·3·5·a·a·a·b·b·b·x·y·y

51 Ex. 3 Simplify each expression. a. 4x 2 y ÷ 2xy 2 15a 3 b 3 5ab 3 4x 2 y ÷ 2xy 2 = 4x 2 y. 5ab 3 15a 3 b 3 5ab 3 15a 3 b 3 2xy 2 = 2·2·5·a·b·b·b·x·x·y 2·3·5·a·a·a·b·b·b·x·y·y

52 Ex. 3 Simplify each expression. a. 4x 2 y ÷ 2xy 2 15a 3 b 3 5ab 3 4x 2 y ÷ 2xy 2 = 4x 2 y. 5ab 3 15a 3 b 3 5ab 3 15a 3 b 3 2xy 2 = 2·2·5·a·b·b·b·x·x·y 2·3·5·a·a·a·b·b·b·x·y·y

53 Ex. 3 Simplify each expression. a. 4x 2 y ÷ 2xy 2 15a 3 b 3 5ab 3 4x 2 y ÷ 2xy 2 = 4x 2 y. 5ab 3 15a 3 b 3 5ab 3 15a 3 b 3 2xy 2 = 2·2·5·a·b·b·b·x·x·y 2·3·5·a·a·a·b·b·b·x·y·y = 2·x 3·a·a·y

54 Ex. 3 Simplify each expression. a. 4x 2 y ÷ 2xy 2 15a 3 b 3 5ab 3 4x 2 y ÷ 2xy 2 = 4x 2 y. 5ab 3 15a 3 b 3 5ab 3 15a 3 b 3 2xy 2 = 2·2·5·a·b·b·b·x·x·y 2·3·5·a·a·a·b·b·b·x·y·y = 2·x = 2x 3·a·a·y 3a 2 y

55 b. x 2 + x – 12 ÷ x + 2 x 2 – 9 x + 3

56 b. x 2 + x – 12 ÷ x + 2 x 2 – 9 x + 3 x 2 + x – 12 ÷ x + 2 = x 2 – 9 x + 3

57 b. x 2 + x – 12 ÷ x + 2 x 2 – 9 x + 3 x 2 + x – 12 ÷ x + 2 = x 2 + x – 12 x 2 – 9 x + 3 x 2 – 9

58 b. x 2 + x – 12 ÷ x + 2 x 2 – 9 x + 3 x 2 + x – 12 ÷ x + 2 = x 2 + x – 12. x + 3 x 2 – 9 x + 3 x 2 – 9 x + 2

59 b. x 2 + x – 12 ÷ x + 2 x 2 – 9 x + 3 x 2 + x – 12 ÷ x + 2 = x 2 + x – 12. x + 3 x 2 – 9 x + 3 x 2 – 9 x + 2 = (x + 4)(x – 3)

60 b. x 2 + x – 12 ÷ x + 2 x 2 – 9 x + 3 x 2 + x – 12 ÷ x + 2 = x 2 + x – 12. x + 3 x 2 – 9 x + 3 x 2 – 9 x + 2 = (x + 4)(x – 3) (x + 3)(x – 3)

61 b. x 2 + x – 12 ÷ x + 2 x 2 – 9 x + 3 x 2 + x – 12 ÷ x + 2 = x 2 + x – 12. x + 3 x 2 – 9 x + 3 x 2 – 9 x + 2 = (x + 4)(x – 3). (x + 3) (x + 3)(x – 3) (x + 2)

62 b. x 2 + x – 12 ÷ x + 2 x 2 – 9 x + 3 x 2 + x – 12 ÷ x + 2 = x 2 + x – 12. x + 3 x 2 – 9 x + 3 x 2 – 9 x + 2 = (x + 4)(x – 3). (x + 3) (x + 3)(x – 3) (x + 2) = (x + 4)(x – 3)(x + 3)

63 b. x 2 + x – 12 ÷ x + 2 x 2 – 9 x + 3 x 2 + x – 12 ÷ x + 2 = x 2 + x – 12. x + 3 x 2 – 9 x + 3 x 2 – 9 x + 2 = (x + 4)(x – 3). (x + 3) (x + 3)(x – 3) (x + 2) = (x + 4)(x – 3)(x + 3) (x + 3)(x – 3)(x + 2)

64 b. x 2 + x – 12 ÷ x + 2 x 2 – 9 x + 3 x 2 + x – 12 ÷ x + 2 = x 2 + x – 12. x + 3 x 2 – 9 x + 3 x 2 – 9 x + 2 = (x + 4)(x – 3). (x + 3) (x + 3)(x – 3) (x + 2) = (x + 4)(x – 3)(x + 3) (x + 3)(x – 3)(x + 2)

65 b. x 2 + x – 12 ÷ x + 2 x 2 – 9 x + 3 x 2 + x – 12 ÷ x + 2 = x 2 + x – 12. x + 3 x 2 – 9 x + 3 x 2 – 9 x + 2 = (x + 4)(x – 3). (x + 3) (x + 3)(x – 3) (x + 2) = (x + 4)(x – 3)(x + 3) (x + 3)(x – 3)(x + 2)

66 b. x 2 + x – 12 ÷ x + 2 x 2 – 9 x + 3 x 2 + x – 12 ÷ x + 2 = x 2 + x – 12. x + 3 x 2 – 9 x + 3 x 2 – 9 x + 2 = (x + 4)(x – 3). (x + 3) (x + 3)(x – 3) (x + 2) = (x + 4)(x – 3)(x + 3) (x + 3)(x – 3)(x + 2)

67 b. x 2 + x – 12 ÷ x + 2 x 2 – 9 x + 3 x 2 + x – 12 ÷ x + 2 = x 2 + x – 12. x + 3 x 2 – 9 x + 3 x 2 – 9 x + 2 = (x + 4)(x – 3). (x + 3) (x + 3)(x – 3) (x + 2) = (x + 4)(x – 3)(x + 3) (x + 3)(x – 3)(x + 2)

68 b. x 2 + x – 12 ÷ x + 2 x 2 – 9 x + 3 x 2 + x – 12 ÷ x + 2 = x 2 + x – 12. x + 3 x 2 – 9 x + 3 x 2 – 9 x + 2 = (x + 4)(x – 3). (x + 3) (x + 3)(x – 3) (x + 2) = (x + 4)(x – 3)(x + 3) (x + 3)(x – 3)(x + 2)

69 b. x 2 + x – 12 ÷ x + 2 x 2 – 9 x + 3 x 2 + x – 12 ÷ x + 2 = x 2 + x – 12. x + 3 x 2 – 9 x + 3 x 2 – 9 x + 2 = (x + 4)(x – 3). (x + 3) (x + 3)(x – 3) (x + 2) = (x + 4)(x – 3)(x + 3) (x + 3)(x – 3)(x + 2) = x + 4 x + 2

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