Chapter 7 Section 7.1- Rational Expressions and...

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Province – Mathematics Department – Southwest Tennessee Community College

Chapter 7 Section 7.1- Rational Expressions and Functions: Multiplying and Dividing Objectives:

1. Define rational expressions. 2. Define rational functions and describe their domains. 3. Write rational expressions in lowest terms. 4. Multiply rational expressions. 5. Find reciprocals for rational expressions. 6. Divide rational expressions.

A relation is a set of ordered pairs of real numbers. The domain, D, of a relation is the set of all first coordinates of the ordered pairs in the relation (the xs). The range, R, of a relation is the set of all second coordinates of the ordered pairs in the relation (the ys). In graphing relations, the horizontal axis is called the domain axis and the vertical axis is called the range axis.

The domain and range of a relation can often be determined from the graph of the relation. **If the domain or range consists of a finite number of points, use braces and set notation. **If the domain or range consists of intervals of real numbers, use interval (or inequality) notation. A rational expression (algebraic fraction) is the quotient of two polynomials, also with the denominator not 0. Rational expressions are the elements of the set

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Define Rational Functions and Describe Their Domain A rational function has the form

The domain of a rational function includes all the real numbers except those that make Q(x), the denominator, equal to 0. **Since division by zero is undefined, the denominator of a fraction CANNOT equal zero. So in order to find the value(s) for which the fraction is not defined (restrictions), just solve the equation DENOMINATOR = 0. The graph of the function f(x) is shown. The domain of this function is all real numbers except x = 3 where f(x) is not defined.

Finding Numbers Not in the Domain of a Rational Function Ex. For the following rational expressions find the restricted values for x. To locate the values not in the domain of a rational function, we need only determine which real numbers make the denominator 0.

86

82

2

xx

xx

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xx

x

92

162

5

152 x

x

16

32

x

x

x

x

20

102 2

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Writing Rational Expressions in Lowest Terms

Simplify the Following Expressions: **To SIMPLIFY an algebraic fraction, 1) Simplify the numerator and denominator applying the

order of operations to each separately, if necessary, in order to help you factor.

2) Factor the numerator and denominator completely. 3) Cancel common pairs of factors if you can by using the

“Cancellation Principle” which states that

if a thenab

ac

b

cbecause

a

a 0 1, , . Remember that

you can also cancel factors that are opposites with a

result of –1 since

a

awhen a1 0.

NOTE: You may cancel only FACTORS of the numerator with FACTORS of the denominator !!!

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Multiplying Rational Expressions

**To MULTIPLY algebraic fractions, 1) Write as one fraction by connecting the fraction bars and

using parentheses if needed, sincea

b

c

d

a c

b d

( )( )

( )( ).

2) Simplify using the method stated above. Examples

3

624 3

3

x

x

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4

3

)3(3

7 m

m

3

4994 2

s

s

st

ss

17

)5(3

5

3 23 rr

r

Finding the Reciprocal of a Rational Expression

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Dividing Rational Expressions

**To DIVIDE algebraic fractions, 1) Change the problem to multiplication by inverting the divisor and changing the division sign to multiplication,

since a

b

c

d

a

b

d

c .

2) Multiply using the method stated above. Examples

5

)3(

15

9 22

xx

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7

)8(4

3

8

aa

n

m

nmn

mm 2

3

652

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Section 7.2 - Adding and Subtracting Rational Expressions Objectives:

1. Add and subtract rational expressions with the same denominator.

2. Find a least common denominator. 3. Add and subtract rational expressions with

different denominators. Adding or Subtracting Rational Expressions Step 1 If the denominators are the same, add or subtract the numerators. Place the result over the common denominator. If the denominators are different, first find the least common denominator. Write all rational expressions with this LCD, and then add or subtract the numerators. Place the result over the common denominator. Step 2 Simplify. Write all answers in lowest terms. **To ADD OR SUBTRACT algebraic fractions, NOTE: You can add or subtract fractions directly only if they have the SAME DENOMINATOR using the following rules:

a

c

b

c

a b

cand

a

c

b

c

a b

c

.

1) Find the least common denominator (LCD). A) Factor each denominator completely. B) Write each in exponential form. C) Circle the highest power of each different base, “prime

factor” that occurs. D) Form a product.

2) Rewrite each fraction in terms of the LCD by multiplying each by a form of 1 so that the value does not change.

a

b

needed factors

needed factors

a needed factors

LCD

.

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3) Substitute these new “equivalent fractions” into the problem, simplifying the

numerators, but leaving the denominators in factored form. 4) Add or subtract across the numerators as indicated and

put over the common denominator. 5) Simplify using the method stated above. Examples

x

x

x

x

8

55

8

48

8

65

8

12

x

x

x

x

Finding the Least Common Denominator Step 1 Factor each denominator. Step 2 Find the least common denominator. The LCD is the product of all different factors from each denominator, with each factor raised to the greatest power that occurs in the denominator.

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Assume that the given expressions are denominators of fractions. Find the LCD for each group. 4m

3n

2, 6m

2n

5

y – 5, y n

2 – 3n – 10, n

2 – 8n + 15

g2 – 2g + 1, g2 + 3g – 4, 5g + 20

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Add or subtract as indicated.

x

y

y

x

69

6

1

4

4

1

x

x

x

x

9

2

3

42

2

x

x

x

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9

3

9

9

xx CAUTION: One of the most common sign errors in algebra occurs when a rational expression with two or more terms in the numerator is being subtracted. In this situation, the subtraction sign must be distributed to every term in the numerator of the fraction that follows it. Carefully study the example below to see how this is done.

Adding and Subtracting Three Rational Expressions

Ex. Add.

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Section 7.3 - Complex Fractions Objectives:

1. Simplify complex fractions by simplifying the numerator and denominator. (Method 1)

2. Simplify complex fractions by multiplying by a common denominator. (Method 2)

3. Compare the two methods of simplifying complex fractions.

4. Simplify rational expressions with negative exponents.

A complex fraction is an expression having a fraction in the numerator, denominator, or both. Examples of complex fractions include

Simplifying a Complex Fraction: Method 1 Step 1 Simplify the numerator and denominator separately. Step 2 Divide by multiplying the numerator by the reciprocal of the denominator. Step 3 Simplify the resulting fraction, if possible. Use Method 1 to simplify each complex fraction

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Simplifying a Complex Fraction: Method 2 Step 1 Multiply the numerator and denominator of the complex

fraction by the least common denominator of the fractions in the numerator and the fractions in the denominator of the complex fraction.

Step 2 Simplify the resulting fraction, if possible.

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Section 7.4 - Equations with Rational Expressions and Graphs Objectives:

1. Determine the domain of the variable in a rational expression.

2. Solve rational equations. Solving Equations Involving Rational Expressions

1. State any restrictions on the variable. 2. Find the LCD of all the fractions. 3. Put parentheses around both sides of the equation, and

multiply both sides by the LCD making sure to distribute where necessary.

4. Solve. 5. CHECK YOUR ANSWERS!!! (Sometimes you get false

solutions.) (Remember that a proportion can be solved by cross multiplication.)

4

5

2

1

4

3

x

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4

3

1

3

x

x

2

3

2

3

1 2

xx

x

xx

x

8

31

2

2

xx

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Solving an Equation with No Solution

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Solving an Equation That Leads to a Quadratic Equation

Chapter 7 Section 7.1- Rational Expressions and...

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