Intermediate Algebra 098A

Chapter 7 Rational Expressions

Intermediate Algebra 098A 7.1

•Introduction • to

•Rational Expressions

Definition: Rational Expression

• Can be written as

• Where P and Q are polynomials and Q(x) is not 0.

Determine domain, range, intercepts

( )( )P xQ x

Determine Domain of rational function.

• 1. Solve the equation Q(x) = 0• 2. Any solution of that

equation is a restricted value and must be excluded from the domain of the function.

Graph

• Determine domain, range, intercepts• Asymptotes

1( )f xx

Graph

• Determine domain, range, intercepts• Asymptotes

2

1( )g xx

Calculator Notes:

• [MODE][dot] useful• Friendly window useful• Asymptotes sometimes occur that are not

part of the graph.• Be sure numerator and denominator are

enclosed in parentheses.

Fundamental Principle of Rational Expressions

ac abc b

Simplifying Rational Expressions to Lowest Terms

• 1. Write the numerator and denominator in factored form.

• 2. Divide out all common factors in the numerator and denominator.

Negative sign rule

p p pq q q

Problem

( 1) 444 1 4

1 41

4

yyy y

yy

Objective:

•Simplify a Rational Expression.

Denise Levertov – U. S. poet

• “Nothing is ever enough. Images split the truth in fractions.”

Robert H. Schuller

• “It takes but one positive thought when given a chance to survive and thrive to overpower an entire army of negative thoughts.”

Intermediate Algebra 098A 7.2

•Multiplication •and

•Division

Multiplication of Rational Expressions

• If a,b,c, and d represent algebraic expressions, where b and d are not 0.

a c acb d bd

Procedure

• 1. Factor each numerator and each denominator completely.

• 2. Divide out common factors.

Procedure

• 1. Factor each numerator and each denominator completely.

• 2. Divide out common factors.

Procedure for Division

• Write down problem• Invert and multiply• Reduce

Objective:

•Multiply and divide rational expressions.

John F. Kennedy – American President

•“Don’t ask ‘why’, ask instead, why not.”

Intermediate Algebra 098A 7.3

•Addition •and

•Subtraction

Objective

• Add and Subtract • rational expressions with

the same denominator.

Procedure adding rational expressions with same

denominator

• 1. Add or subtract the numerators

• 2. Keep the same denominator.• 3. Simplify to lowest terms.

Algebraic Definition

a b a bc c ca b a bc c c

Intermediate Algebra 098A 7.4

• Adding and Subtracting Rational Expressions with unlike Denominators

LCMLCD

• The LCM – least common multiple of denominators is called LCD – least common denominator.

Objective

• Find the lest common denominator (LCD)

Determine LCM of polynomials

• 1. Factor each polynomial completely – write the result in exponential form.

• 2. Include in the LCM each factor that appears in at least one polynomial.

• 3. For each factor, use the largest exponent that appears on that factor in any polynomial.

Procedure: Add or subtract rational expressions with different denominators.

• 1. Find the LCD and write down• 2. “Build” each rational expression so

the LCD is the denominator.• 3. Add or subtract the numerators and

keep the LCD as the denominator.• 4. Simplify

Elementary Example

• LCD = 2 x 3

1 2 1 3 2 22 3 2 3 3 2

3 4 3 4 76 6 6 6

Objective

• Add and Subtract • rational expressions with

unlike denominator.

Martin Luther

• “Even if I knew that tomorrow the world would go to pieces, I would still plant my apple tree.”

Maya Angelou - poet

• “Since time is the one immaterial object which we cannot influence – neither speed up nor slow down, add to nor diminish – it is an imponderably valuable gift.”

Intermediate Algebra 098A 7.5

•Equations •with

•Rational Expressions

Extraneous Solution

• An apparent solution that is a restricted value.

Procedure to solve equations containing rational expressions

• 1. Determine and write LCD• 2. Eliminate the denominators of the

rational expressions by multiplying both sides of the equation by the LCD.

• 3. Solve the resulting equation• 4. Check all solutions in original

equation being careful of extraneous solutions.

Graphical solution

• 1. Set = 0 , graph and look for x intercepts.• Or• 2. Graph left and right sides and look for

intersection of both graphs.• Useful to check for extraneous solutions

and decimal approximations.

Thomas Carlyle

•“Ever noble work is at first impossible.”

Intermediate Algebra 098A 7.6

• Applications• Proportions and Problem

Solving• With

• Rational Equations

Objective

• Use Problem Solving methods including charts, and table to solve problems with two unknowns involving rational expressions.

Problems involving work

• (person’s rate of work) x (person's time at work) = amount of the task completed by that person.

Work problems continued

• (amount completed by one person) + (amount completed by the other person) = whole task

Intermediate Algebra 098A 7.7

• Simplifying Complex Fractions

Definition: Complex rational expression

• Is a rational expression that contains rational expressions in the numerator and denominator.

Objective

• Simplify a complex rational expression.

Procedure 1

• 1. Simplify the numerator and denominator if needed.

• 2. Rewrite as a horizontal division problem.

• 3. Invert and multiply• Note – works best when fraction over

fraction.

Procedure 2

• 1. Multiply the numerator and denominator of the complex rational expression by the LCD of the secondary denominators.

• 2. Simplify• Note: Best with more complicated

expressions.• Be careful using parentheses where

needed.

Paul J. Meyer

• “Enter every activity without giving mental recognition to the possibility of defeat. Concentrate on your strengths, instead of your weaknesses…on your powers, instead of your problems.”