intermediate algebra chapter 6 - gay

Intermediate Algebra Chapter 6 - Gay

•Rational Expressions

Intermediate Algebra 6.1

•Introduction

• to


Definition: Rational Expression

• Can be written as

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

Determine domain, range, intercepts

( )

( )

P x

Q 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 x

x

Graph

• Determine domain, range, intercepts

• Asymptotes

2

1( )g x

x

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 a

bc 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 p

q q q

Problem

( 1) 44

4 1 4

1 41

4

yy

y y

y

y

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.”


•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 ac

b d bd

Procedure

• 1. Factor each numerator and each denominator completely.

• 2. Divide out common factors.

Definition of Division of Rational Expressions

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

a c a d ad

b d b c bc

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.”


•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 b

c c ca b a b

c c c

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 2

2 3 2 3 3 2

3 4 3 4 7

6 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.”


•Complex Fractions

Definition: Complex rational expression

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

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.

Objective

• Simplify a complex rational expression.

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.”


•Division

Long division Problems

2 5 7

2

x x

x

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.”


•Equations

•with


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.

Proportions and Cross Products

• If

, 0a c

where b db dthen ad bc

Thomas Carlyle

•“Ever noble work is at first impossible.”


•Applications

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

Section 6.7 – GayVariation and Problem Solving

• Direct Variation

• Inverse Variation

• Joint Variation

• Applications

Def: Direct Variation

• The value of y varies directly with the value of x if there is a constant k such that y = kx.

Objective

• Solve Direct Variation Problems

• Determine constant of proportionality.

Procedure:Solving Variation Problems

• 1. Write the equation • Example y = kx• 2. Substitute the initial values and

find k.• 3. Substitute for k in the original

equation• 4. Solve for unknown using new

equation.

Example: Direct Variation

• y varies directly as x. If y = 18 when x = 5, find y when x = 8

• Answer: y = 28.8

Helen Keller – advocate for he blind

•“Alone we can do so little, together we can do so much.”

Definition: Inverse Variation

• A quantity y varies inversely with x if there is a constant k such that

• y is inversely proportional to x.

• k is called the constant of variation.

ky

x

Procedure: Solving inverse variation problems

• 1. Write the equation• 2. Substitute the initial values

and find k• 3. Substitute for k in the

equation found in step 1.• 4. Solve for the unknown.

Joint Variation

• Three variables y,x,z are in joint variation if y = kxz where k is a constant.

Leonardo Da Vinci - scientist, inventor, and artist

•“Time stays long enough for those who use it.”

intermediate algebra chapter 6 - gay

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