Master College Algebra: Rational Expressions and Domain

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

2. Prerequisites P

3. Rational Expressions P.8

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

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

6. 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 domainof 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 forx = 2. • Find the domain of the expression.

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

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

15. Example (b) E.g. 2—Finding the Domain • 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. Example (c) E.g. 2—Finding the Domain • 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 cancelcommon factors from the numerator and denominator.

19. E.g. 3—Simplifying by Cancellation • Simplify:

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

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

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

37. Compound Fractions

38. Compound Fraction • A compound fractionis: • A fraction in which the numerator, the denominator, or both, are themselves fractional expressions.

39. E.g. 7—Simplifying a Compound Fraction • Simplify:

40. Solution 1 E.g. 7—Simplifying • 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. Solution 1 E.g. 7—Simplifying • Thus,

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

43. Solution 2 E.g. 7—Simplifying • 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 • Simplify: • 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 • Simplify:

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

50. Solution 2 E.g. 9—Simplifying • (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.