1. MATH 31 LESSONS PreCalculus 3. Simplifying Rational Expressions Rationalization

2. A. Rational Expressions To simplify a rational expression:  Factor everything in the numerator and denominator  Cancel any shared factor between the top and the bottom  Leave the answer in factored form

3. e.g.Simplify

4. Factor the top and the bottom

5. Be certain they are fully factored

6. Cancel any shared factor between the top and bottom

7. Multiplying Rational Expressions Let’s start by reviewing how to multiply fractions. Try multiplying

8. When you multiply fractions, you multiply: top  top bottom  bottom

10. Now we will try multiplying rational expressions. Multiply and simplify

11. Fully factor all expressions

12. Multiply: top  top, bottom  bottom Note: This is an optional step. You don’t have to do it.

13. Cancel any shared factors between the top and bottom

14. Dividing Rational Expressions Let’s start by reviewing how to divide fractions. Try dividing

15. When you divide a fraction, you “invert and multiply”

16. When you multiply fractions, it is: top  top bottom  bottom

17. Now we will try dividing rational expressions. Divide and simplify

18. When you divide a fraction, you “invert and multiply”

19. Fully factor all expressions

20. There is no need to multiply them. Simply cancel any shared factors on the top and bottom.

21. Adding and Subtracting Rational Expressions Let’s start by reviewing how to add and subtract fractions. Try simplifying

22. LCD: 12 Multiply the top and bottom of each fraction by the number which will make the LCD

24. When they have the same denominator, you add / subtract the numerators (while keeping the denominator the same)

26. Adding and Subtracting Rational Expressions For rational expressions, use the following procedure:  Find the LCD - you may need to factor the denominators first - the LCD of the variables is the largest power  For each fraction, multiply the top and bottom with the “missing factor” to create the LCD  Add or subtract the numerator, while leaving the denominator the same

27. e.g. 1Simplify

28. LCD: 24x 2 y Multiply the top and bottom of each fraction with what’s needed to make the LCD

30. When the denominator is the same, you can add the numerators (while leaving the denominator the same)

31. e.g. 2Simplify

32. Factor the denominators

33. LCD: x (x - 3) (x + 1 Multiply the top and bottom of each fraction with the “missing factors”

34. Since the denominators are the same, you can subtract the numerators (and keep the denominator the same)

35. Simplify the numerator

36. Keep the denominator factored

37. Ex. 1Simplify Try this example on your own first. Then, check out the solution.

38. When you divide a fraction, invert and multiply. Also, put the polynomials in the same order (x’s first, constant last)

39. Factor all expressions

41. Cancel any shared factors between the numerator and the denominator

44. Ex. 2Simplify Try this example on your own first. Then, check out the solution.

45. Factor the denominators

46. LCD: x (x - 10) (x - 5) Multiply the top and bottom of the fractions with the “missing factors”

47. When the denominators are the same, simply subtract the numerators (leaving the denominator the same)

48. Simplify the numerator, but leave the denominator factored

49. Don’t stop yet. What else can be done?

52. Ex. 3Simplify Try this example on your own first. Then, check out the solution.

53. Bring the fractions together (using LCD) on top and bottom

55. When you divide by a fraction, invert and multiply

58. After factoring, there are no shared factors. It cannot be simplified further.

59. B. Rationalizing Radical Expressions B1. Radical Arithmetic If a > 0,

60. Squaring and square-rooting are “opposites” (inverses), and so, they cancel each other out

61. e.g. Express as a simplified mixed radical

62. Factor out the largest perfect square in the radicand

63. Separate the radicals

66. Radical Arithmetic If c > 0,

67. If they are like radicals, you can add (or subtract) the coefficients. The radical stays the same.

68. e.g. Simplify

71. Radical Arithmetic If a, b > 0,

72. When you multiply radicals, you simply multiply the radicands (the numbers inside the radical)

73. If a, b > 0, Similarly,

74. e.g. Simplify

79. Ex. 4Simplify Try this example on your own first. Then, check out the solution.

80. It is recommended that you write it out twice. Remember: (a + b) 2 ≠ a 2 + b 2

81. Use FOIL to expand.

83. Bring together like radicals

86. B2. Rationalizing Denominators When you rationalize a denominator, you are removing all radicals from the denominator. We will consider two cases: 1. Monomial denominators 2. Binomial denominators

87. Case 1: Monomial denominators Use the following procedure:  Simplify the radicals as much as possible - no perfect squares in the radicand  Multiply the top and bottom by the radical in the denominator

88. e.g. Rationalize the denominator for

89. Simplify all radicals first

90. Multiply the top and bottom by the radical in the denominator. No need to multiply by the coefficient, since it is already rational

91. If rationalization is done properly, the radical becomes squared

92. Now, there are no radicals in the denominator. It has been rationalized.

94. Case 2: Binomial denominators Use the following procedure:  Multiply the top and bottom by the “conjugate” of the denominator “a + b” and “a  b” are called conjugates  This should result in a difference of squares (a 2 - b 2 ), which removes the radicals

95. e.g. 1 Rationalize the denominator for

96. Multiply the top and bottom by the “conjugate” of the denominator. If the denominator is of the form “a - b”, then the conjugate is “a + b”

97. If done properly, each of the radicals in the denominator will be squared. (a + b) (a - b) = a 2 - b 2

98. Now, there are no radicals in the denominator. It has been rationalized.

100. e.g. 2 Rationalize the denominator for

101. Multiply the top and bottom by the conjugate of the denominator. If the denominator is of the form “a + b”, then the conjugate is “a - b”

102. This results in the radical being squared. (a + b) (a - b) = a 2 - b 2

103. There are no radicals in the denominator. It has been rationalized.

104. Simplify the denominator, but leave the numerator factored. There may be an opportunity to simplify.

106. Ex. 5Rationalize the denominator for Try this example on your own first. Then, check out the solution.

107. Multiply the top and bottom by the conjugate. If the denominator is of the form “a + b”, then the conjugate is “a - b”

108. The radical in the denominator should become squared. (a + b) (a - b) = a 2 - b 2

109. There are no radicals in the denominator. It has been rationalized.

110. Leave the top factored. There may be an opportunity to simplify.

112. Ex. 5Rationalize the numerator for Try this example on your own first. Then, check out the solution.

113. Multiply the top and bottom by the conjugate of the numerator. If the numerator is of the form “a - b”, then the conjugate is “a + b”

114. If done properly, the radical in the numerator will be squared. (a - b) (a + b) = a 2 - b 2

115. Do not simplify the denominator. There may be an opportunity to simplify.

119. Ex. 6Rationalize the numerator for Try this example on your own first. Then, check out the solution.

120. Multiply the top and bottom by the conjugate of the numerator. If the numerator is of the form “a - b”, then the conjugate is “a + b”

121. If done properly, the radicals will become squared. (a - b) (a + b) = a 2 - b 2

122. The numerator no longer has a radical. It has been rationalized.

123. Do not simplify the denominator. There may be an opportunity to cancel.