Intermediate Algebra Chapter 6 Gay Rational Expressions Intermediate

Intermediate Algebra Chapter 6 - Gay • Rational Expressions

Intermediate Algebra 6. 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 of rational function. 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. • 1.

Graph • Determine domain, range, intercepts • Asymptotes

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

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

Problem

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 6. 1 • Multiplication • and • Division

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

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

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 6. 2 • Addition • and • Subtraction

Objective • Add and Subtract • rational expressions with the same denominator.

Procedure adding rational expressions with same denominator Add or subtract the numerators • 2. Keep the same denominator. • 3. Simplify to lowest terms. • 1.

Algebraic Definition

LCM LCD • 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

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. ”

Intermediate Algebra 6. 3 • 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. ”

Intermediate Algebra 6. 4 • Division

Long division Problems

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 6. 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.

Proportions and Cross Products • If

Thomas Carlyle • “Ever noble work is at first impossible. ”

Intermediate Algebra 6. 6 • 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 – Gay Variation 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.

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. ”