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College Algebra

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Title: College Algebra


1
  • College Algebra
  • Sixth Edition
  • James Stewart ? Lothar Redlin ? Saleem Watson

2
P
  • Prerequisites

3
P.7
  • 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
    in which 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 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. 1Finding the Domain of an Expression
  • Find the domains of these expressions.

11
E.g. 1Finding the Domain
Example (a)
  • 2x2 3x 1
  • This polynomial is defined for every x.
  • Thus, the domain is the set of real numbers.

12
E.g. 1Finding the Domain
Example (b)
  • We first factor the denominator
  • Since the denominator is zero when x 2 or x
    3.
  • The expression is not defined for these numbers.
  • The domain is x x ? 2 and x ? 3.

13
E.g. 1Finding 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.

14
  • Simplifying Rational Expressions

15
Simplifying Rational Expressions
  • To simplify rational expressions, we factor both
    numerator and denominator and use following
    property of fractions
  • This allows us to cancel common factors from the
    numerator and denominator.

16
E.g. 2Simplifying Rational Expressions by
Cancellation
  • Simplify

17
Caution
  • We cant cancel the x2s in
  • because x2 is not a factor.

18
  • Multiplying and Dividing Rational Expressions

19
Multiplying Rational Expressions
  • To multiply rational expressions, we use the
    following property of fractions
  • This says that
  • To multiply two fractions, we multiply their
    numerators and multiply their denominators.

20
E.g. 3Multiplying Rational Expressions
  • Perform the indicated multiplication, and
    simplify

21
E.g. 3Multiplying Rational Expressions
  • We first factor

22
Dividing Rational Expressions
  • To divide rational expressions, we use the
    following property of fractions
  • This says that
  • To divide a fraction by another fraction, we
    invert the divisor and multiply.

23
E.g. 4Dividing Rational Expressions
  • Perform the indicated division, and simplify

24
E.g. 4Dividing Rational Expressions

25
  • Adding and Subtracting Rational Expressions

26
Adding and Subtracting Rational Expressions
  • To add or subtract rational expressions, we
    first find a common denominator and then use the
    following property of fractions

27
Adding and Subtracting Rational Expressions
  • Although any common denominator will work, it is
    best to use the least common denominator (LCD) as
    explained 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.

28
Caution
  • Avoid making the following error

29
Caution
  • For instance, if we let A 2, B 1, and C 1,
    then we see the error

30
E.g. 5Adding and Subtracting Rational Expressions
  • Perform the indicated operations, and simplify

31
E.g. 5Adding Rational Exp.
Example (a)

Here LCD is simply the product (x 1)(x 2).
32
E.g. 5Subtracting Rational Exp.
Example (b)
  • The LCD of x2 1 (x 1)(x 1) and (x 1)2
    is (x 1)(x 1)2.

33
E.g. 5Subtracting Rational Exp.
Example (b)

34
  • Compound Fractions

35
Compound Fraction
  • A compound fraction is
  • A fraction in which the numerator, the
    denominator, or both, are themselves fractional
    expressions.

36
E.g. 6Simplifying a Compound Fraction
  • Simplify

37
E.g. 6Simplifying
Solution 1
  • One solution is as follows.
  • We combine the terms in the numerator into a
    single fraction.
  • We do the same in the denominator.
  • Then we invert and multiply.

38
E.g. 6Simplifying
Solution 1
  • Thus,

39
E.g. 6Simplifying
Solution 2
  • Another solution is as follows.
  • We find the LCD of all the fractions in the
    expression,
  • Then multiply the numerator and denominator by
    it.

40
E.g. 6Simplifying
Solution 2
  • Here, the LCD of all the fractions is xy.

41
Simplifying a Compound Fraction
  • The next two examples show situations in calculus
    that require the ability to work with fractional
    expressions.

42
E.g. 7Simplifying a Compound Fraction
  • Simplify
  • We begin by combining the fractions in the
    numerator using a common denominator

43
E.g. 7Simplifying a Compound Fraction
44
E.g. 7Simplifying a Compound Fraction

45
E.g. 8Simplifying a Compound Fraction
  • Simplify

46
E.g. 8Simplifying
Solution 1
  • Factor (1 x2)1/2 from the numerator.

47
E.g. 8Simplifying
Solution 2
  • Since (1 x2)1/2 1/(1 x2)1/2 is a fraction,
  • we can clear all fractions by multiplying
    numerator and denominator by (1 x2)1/2.

48
E.g. 8Simplifying
Solution 2
Thus,

49
  • Rationalizing the Denominator or the Numerator

50
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 .

51
Rationalizing the Denominator
  • This is effective because, by Product Formula 1
    in Section P.5, the product of the denominator
    and its conjugate radical does not contain a
    radical

52
E.g. 9Rationalizing the Denominator
  • Rationalize the denominator
  • We multiply both the numerator and the
    denominator by the conjugate radical of ,
    which is .

53
E.g. 9Rationalizing the Denominator
  • Thus,

54
E.g. 10Rationalizing the Numerator
  • Rationalize the numerator
  • We multiply numerator and denominator by the
    conjugate radical

55
E.g. 10Rationalizing the Numerator
  • Thus,

56
E.g. 10Rationalizing the Numerator

57
  • Avoiding Common Errors

58
Avoiding Common Errors
  • Dont make the mistake of applying properties of
    multiplication to the operation of addition.
  • Many of the common errors in algebra involve
    doing just that.

59
Avoiding Common Errors
  • The following table states several multiplication
    properties and illustrates the error in applying
    them to addition.

60
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.

61
Avoiding Common Errors
  • For example, if we take a 2 and b 2 in the
    fourth error, we get different values for the
    left- and right-hand sides

62
Avoiding Common Errors
  • The left-hand side is
  • The right-hand side is
  • Since 1 ? ¼, the stated equation is wrong.

63
Avoiding Common Errors
  • You should similarly convince yourself of the
    error in each of the other equations.
  • See Exercise 119.
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