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CHAPTER 7 Rational Expressions and Equations
Imagine you and a friend have partnered in a business venture. You have opened a computer service shop in a local mall. The profits are good, but they seem to vary depending on how much advertising you do for the business. Do you
7.1
Simplifying Rational Expressions
431
7.2
Multiplying and Dividing Rational Expressions
437
7.3
Adding and Subtracting Rational Expressions
442
How Am I Doing? Sections 7.1–7.3
450
7.4
Simplifying Complex Rational Expressions
451
7.5
Solving Equations Involving Rational Expressions
457
7.6
Ratio, Proportion, and Other Applied Problems
462
Use Math to Save Money
470
Chapter 7 Organizer
471
Chapter 7 Review Problems
472
How Am I Doing? Chapter 7 Test
476
Math Coach
477
430431
7.1 Simplifying Rational Expressions
Student Learning Objective
After studying this section, you will be able to:
Simplify rational expressions by factoring.
Recall that a rational number is a number that can be written as one integer divided by another integer, such as 3 ÷ 4 or
. We usually use the word
fraction
to mean
. We can extend this idea to algebraic expressions. A
rational exp
The last fraction is sometimes also called a
fractional algebraic expression.
There is a special restriction for all fractions, including fractional algebraic expressions: The denominator of the fraction cannot be 0. For example, in the ex
the denominator cannot be 0. Therefore,
the value of
x
cannot be
−
4.
The following important restriction will apply throughout this chapter. We state it here to avoid having to mention it repeatedly throughout this chapter.
RESTRICTION
The denominator of a rational expression cannot be zero. Any value of the variable that would make the denominator zero is not allowed.
We have discovered that fractions can be simplified (or reduced) in the following way.
This is sometimes referred to as the
basic rule of fractions
and can be stated as follows.
BASIC RULE OF FRACTIONS
For any rational expression
and any polynomials
a, b
, and
c
(where
b
≠
0 and
c
≠
0),
We will examine several examples where
a, b
, and
c
are real numbers, as well as more involved examples where
a, b
, and
c
are polynomials. In either case we shall make extensive use of our factoring skills in this section.
One essential property is revealed by the basic rule of fractions: If the numerator and denominator of a given fraction are multiplied by the same nonzero quantity, an equivalent fraction is obtained. The rule can be used two ways. You
EXAMPLE 1
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- Rational Expressions, Fractions, Fraction, Elementary arithmetic, Greatest common divisor