1. College Algebra Fifth Edition
James Stewart  Lothar Redlin  Saleem Watson

2. P
Prerequisites

3. P.8
Rational Expressions

4. Fractional Expression
A quotient of two algebraic expressions is called a fractional expression.
Here are some examples:

5. Rational Expression
A rational expression is a fractional expression where both the numerator and denominator are polynomials.
Here are some examples:

6. In this section, we learn:
Rational Expressions
In this section, we learn:
How to perform algebraic operations on rational expressions.

7. The Domain of an Algebraic Expression

8. The Domain of an Algebraic Expression
In general, an algebraic expression may not be defined for all values of the variable.
The domain of an algebraic expression is:
The set of real numbers that the variable is permitted to have.

9. The Domain of an Algebraic Expression
The table gives some basic expressions and their domains.

10. E.g. 1—Finding the Domain of an Expression
Consider the expression
Find the value of the expression for x = 2.
Find the domain of the expression.

11. E.g. 1—The Value of an Expression
Example (a)
We find the value by substituting 2 for x in the expression:

12. E.g. 1—Domain of an Expression
Example (b)
The denominator is zero when x = 3.
Since division by zero is not defined:
We have x ≠ 3.
Thus, the domain is all real numbers except 3.
We can write this in set notation as {x | x ≠ 3}

13. E.g. 2—Finding the Domain of an Expression
Find the domains of these expressions.

14. This polynomial is defined for every x.
E.g. 2—Finding the Domain
Example (a)
2x2 + 3x – 1
This polynomial is defined for every x.
Thus, the domain is the set of real numbers.

15. We first factor the denominator.
E.g. 2—Finding the Domain
Example (b)
We first factor the denominator.
The denominator is zero when x = 2 or x = 3.
So, the expression is not defined for these numbers.
Hence, the domain is: {x | x ≠ 2 and x ≠ 3}

16. For the numerator to be defined, we must have x ≥ 0.
E.g. 2—Finding the Domain
Example (c)
For the numerator to be defined, we must have x ≥ 0.
Also, we cannot divide by zero; so, x ≠ 5.
Thus, the domain is: {x | x ≥ 0 and x ≠ 5}

17. Simplifying Rational Expressions

18. Simplifying Rational Expressions
To simplify rational expressions, we factor both numerator and denominator and use this property of fractions:
This allows us to cancel common factors from the numerator and denominator.

19. E.g. 3—Simplifying by Cancellation

20. We can’t cancel the x2’s in
Caution
We can’t cancel the x2’s in
because x2 is not a factor.

21. Multiplying and Dividing Rational Expressions

22. Multiplying Rational Expressions
To multiply rational expressions, we use this property of fractions:
This says that:
To multiply two fractions, we multiply their numerators and multiply their denominators.

23. E.g. 4—Multiplying Rational Expressions
Perform the indicated multiplication and simplify:

24. E.g. 4—Multiplying Rational Expressions
We first factor.

25. Dividing Rational Expressions
To divide rational expressions, we use this property of fractions:
This says that:
To divide a fraction by another fraction, we invert the divisor and multiply.

26. E.g. 5—Dividing Rational Expressions
Perform the indicated division and simplify:

27. E.g. 5—Dividing Rational Expressions

28. Adding and Subtracting Rational Expressions

29. Adding and Subtracting Rational Expressions
To add or subtract rational expressions, we first find a common denominator and then use this property of fractions:

30. Adding and Subtracting Rational Expressions
Any common denominator will work.
Still, it is best to use the least common denominator (LCD) as learnt in Section P.2.
The LCD is found by factoring each denominator and taking the product of the distinct factors, using the highest power that appears in any of the factors.

31. Avoid making the following error:
Caution
Avoid making the following error:

32. Caution
For instance, if we let A = 2, B = 1, and C = 1, then we see the error:

33. E.g. 6—Adding and Subtracting Rational Expressions
Perform the indicated operations and simplify:

34. The LCD is simply the product (x – 1)(x + 2).
E.g. 6—Adding
Example (a)
The LCD is simply the product (x – 1)(x + 2).

35. The LCD of x2 – 1 = (x – 1)(x + 1) and (x + 1)2 is (x – 1)(x + 1)2.
E.g. 6—Subtracting
Example (b)
The LCD of x2 – 1 = (x – 1)(x + 1) and (x + 1)2 is (x – 1)(x + 1)2.

36. E.g. 6—Subtracting Rational Exp.
Example (b)

37. Compound Fractions

38. A compound fraction is:
A fraction in which the numerator, the denominator, or both, are themselves fractional expressions.

39. E.g. 7—Simplifying a Compound Fraction

40. One solution is as follows.
E.g. 7—Simplifying
Solution 1
One solution is as follows.
Combine the terms in the numerator into a single fraction.
Do the same in the denominator.
Invert and multiply.

41. E.g. 7—Simplifying
Solution 1
Thus,

42. Another solution is as follows.
E.g. 7—Simplifying
Solution 2
Another solution is as follows.
Find the LCD of all the fractions in the expression.
Multiply the numerator and denominator by it.

43. Here, the LCD of all the fractions is xy.
E.g. 7—Simplifying
Solution 2
Here, the LCD of all the fractions is xy.

44. Simplifying a Compound Fraction
The next two examples show situations in calculus that require the ability to work with fractional expressions.

45. E.g. 8—Simplifying a Compound Fraction
We begin by combining the fractions in the numerator using a common denominator.

46. E.g. 8—Simplifying a Compound Fraction

47. E.g. 8—Simplifying a Compound Fraction

48. E.g. 9—Simplifying a Compound Fraction

49. Factor (1 + x2)–1/2 from the numerator.
E.g. 9—Simplifying
Solution 1
Factor (1 + x2)–1/2 from the numerator.

50. (1 + x2)–1/2 = 1/(1 + x2)1/2 is a fraction.
E.g. 9—Simplifying
Solution 2
(1 + x2)–1/2 = 1/(1 + x2)1/2 is a fraction.
Therefore, we can clear all fractions by multiplying numerator and denominator by (1 + x2)1/2.

51. E.g. 9—Simplifying
Solution 2
Thus,

52. Rationalizing the Denominator or the Numerator

53. Rationalizing the Denominator
If a fraction has a denominator of the form we may rationalize the denominator by multiplying numerator and denominator by the conjugate radical

54. Rationalizing the Denominator
This is effective because, by Special Product Formula 1 in Section P.6, the product of the denominator and its conjugate radical does not contain a radical:

55. E.g. 10—Rationalizing the Denominator
Rationalize the denominator:
We multiply both the numerator and the denominator by the conjugate radical of , which is

56. E.g. 10—Rationalizing the Denominator
Thus,

57. E.g. 11—Rationalizing the Numerator
Rationalize the numerator:
We multiply numerator and denominator by the conjugate radical

58. E.g. 11—Rationalizing the Numerator
Thus,

59. E.g. 10—Rationalizing the Numerator

60. Avoiding Common Errors

61. Avoiding Common Errors
Don’t make the mistake of applying properties of multiplication to the operation of addition.
Many of the common errors in algebra involve doing just that.

62. Avoiding Common Errors
The table states several multiplication properties and illustrates the error in applying them to addition.

63. Avoiding Common Errors
To verify that the equations in the right-hand column are wrong, simply substitute numbers for a and b and calculate each side.

64. Avoiding Common Errors
For example, if we take a = 2 and b = 2 in the fourth error, we have the following result.

65. Avoiding Common Errors
The left-hand side is:
The right-hand side is:
Since 1 ≠ ¼, the stated equation is wrong.

66. Avoiding Common Errors
You should similarly convince yourself of the error in each of the other equations.
See Exercise 113.