Title: College Algebra
1- College Algebra
- Sixth Edition
- James Stewart ? Lothar Redlin ? Saleem Watson
2P
3P.7
4Fractional Expression
- A quotient of two algebraic expressions is called
a fractional expression. - Here are some examples
5Rational Expression
- A rational expression is a fractional expression
in which both the numerator and denominator are
polynomials. - Here are some examples
6Rational Expressions
- In this section, we learn
- How to perform algebraic operations on rational
expressions.
7- The Domain of an Algebraic Expression
8The 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.
9The Domain of an Algebraic Expression
- The table gives some basic expressions and their
domains.
10E.g. 1Finding the Domain of an Expression
- Find the domains of these expressions.
11E.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.
12E.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.
13E.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
15Simplifying 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.
16E.g. 2Simplifying Rational Expressions by
Cancellation
17Caution
- We cant cancel the x2s in
- because x2 is not a factor.
18- Multiplying and Dividing Rational Expressions
19Multiplying 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.
20E.g. 3Multiplying Rational Expressions
- Perform the indicated multiplication, and
simplify
21E.g. 3Multiplying Rational Expressions
22Dividing 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.
23E.g. 4Dividing Rational Expressions
- Perform the indicated division, and simplify
24E.g. 4Dividing Rational Expressions
25- Adding and Subtracting Rational Expressions
26Adding and Subtracting Rational Expressions
- To add or subtract rational expressions, we
first find a common denominator and then use the
following property of fractions
27Adding 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.
28Caution
- Avoid making the following error
29Caution
- For instance, if we let A 2, B 1, and C 1,
then we see the error
30E.g. 5Adding and Subtracting Rational Expressions
- Perform the indicated operations, and simplify
31E.g. 5Adding Rational Exp.
Example (a)
Here LCD is simply the product (x 1)(x 2).
32E.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.
33E.g. 5Subtracting Rational Exp.
Example (b)
34 35Compound Fraction
- A compound fraction is
- A fraction in which the numerator, the
denominator, or both, are themselves fractional
expressions.
36E.g. 6Simplifying a Compound Fraction
37E.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.
38E.g. 6Simplifying
Solution 1
39E.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.
40E.g. 6Simplifying
Solution 2
- Here, the LCD of all the fractions is xy.
41Simplifying a Compound Fraction
- The next two examples show situations in calculus
that require the ability to work with fractional
expressions.
42E.g. 7Simplifying a Compound Fraction
- Simplify
- We begin by combining the fractions in the
numerator using a common denominator
43E.g. 7Simplifying a Compound Fraction
44E.g. 7Simplifying a Compound Fraction
45E.g. 8Simplifying a Compound Fraction
46E.g. 8Simplifying
Solution 1
- Factor (1 x2)1/2 from the numerator.
47E.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.
48E.g. 8Simplifying
Solution 2
Thus,
49- Rationalizing the Denominator or the Numerator
50Rationalizing 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 .
51Rationalizing 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
52E.g. 9Rationalizing the Denominator
- Rationalize the denominator
- We multiply both the numerator and the
denominator by the conjugate radical of ,
which is .
53E.g. 9Rationalizing the Denominator
54E.g. 10Rationalizing the Numerator
- Rationalize the numerator
- We multiply numerator and denominator by the
conjugate radical
55E.g. 10Rationalizing the Numerator
56E.g. 10Rationalizing the Numerator
57 58Avoiding 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.
59Avoiding Common Errors
- The following table states several multiplication
properties and illustrates the error in applying
them to addition.
60Avoiding 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.
61Avoiding 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
62Avoiding Common Errors
- The left-hand side is
- The right-hand side is
- Since 1 ? ¼, the stated equation is wrong.
63Avoiding Common Errors
- You should similarly convince yourself of the
error in each of the other equations. - See Exercise 119.