Math 072 Notes Chapter 5 PDF

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This is a document of chapter 5 lecture notes....

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5.1 - Rational Expressions and Functions; Multiplying, Dividing, and Simplifying Pg1 Key ideas: • Be able to recognize a rational expression/function, and determine its domain. • Know how to simply rational expressions. • Understand how to multiply, divide, and simplify rational expressions.

Continuing our practice with polynomials from chapter 4, we now focus on rational expressions. Rational Expressions - (Try to recall the definition of rational numbers) Definition

is a rational expression if:

Examples of rational expressions are:

Finding the Domain or Restricted Values Example 1. Find the restricted values for (if there are any) using the rational expression given below. Give the domain in interval notation. a.)

Simplifying Rational Expressions - Simplifying rational expressions means removing any common factors of both the numerator and denominator. To do this, start by factoring the numerator and denominator! Example 2. Simplify. a.)

b.)

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c.)

5.1 - Rational Expressions and Functions; Multiplying, Dividing, and Simplifying Pg2 Multiplying Rational Expressions To multiply rational expressions, recall how we multiply fractions in general. Example 3. Multiply and simplify.

a.)

Dividing Rational Expressions

b.)

To divide rational expressions, recall how we divide fractions in general.

Example 4. Divide and simplify.

a.)

b.)

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5.2 - LCMs, LCDs, Addition and Subtraction Pg 1 Key ideas: • Be able to find the LCM of an expression by factoring. • Know how to add or subtract rational expressions and simplify the result.

Review: Adding and Subtracting Fractions by finding the least common denominator. Example 1. Find the least common denominator, and then add or subtract. b.)

a.)

LCD =

c.)

LCD =

LCD =

Adding or Subtracting Rational Expressions by Finding the LCD. Example 2. Add or subtract the following rational expressions. Be sure to find the LCD. a.)

LCD =

c.)

LCD =

b.)

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LCD =

5.2 - LCMs, LCDs, Addition and Subtraction Pg 2 Example 2 continued. Add or subtract the following rational expressions. Be sure to find the LCD. d.)

LCD =

e.)

LCD =

f.)

LCD =

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5.3 - Division by Monomials Key ideas: • Divide Polynomials by Monomials • No Synthetic Division or Polynomial Long Division

Main Idea - When we divide a polynomial by a monomial, we can write this as the sum of each term divided by that monomial. See below.

Example 1. Divide each and simplify. a.)

b.)

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5.4 - Complex Rational Expressions Key ideas: • Identify Complex Rational Expressions. • Simplify Complex Rational Expressions. • Recognize fully simplified complex rational expressions. Identifying "Complex" Rational Expressions A complex rational expression is a ratio in which the numerator and/or denominator itself contains a rational expression (at least one of which the denominator is not 1.). See the examples below. Consider the expression:

. Now consider the following substitutions for and .

then

and

and

then

Simplifying Complex Rational Expressions To simplify a complex rational expression we do the following: ○ Rewrite it as a rational expression through division ○ Simplify that rational expression by removing common factors. Example 1. Simplify each.

a.)

d.)

b.)

c.)

e.)

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5.5 Rational Equations Pg1

Key ideas: • Be able to recognize restricted solutions. • Know the difference between rational equations and rational expressions. • Understand how to solve rational equations and be able to execute the process.

A Method for Solving Rational Equations 1.) Look to factor each denominator completely. 2.) Determine the value(s) of the variables that would make each denominator zero. Exclude these as possible solutions. Make a note somewhere so you remember these values. 3.) Determine of the LCM of all the denominators in the problem. 4.) CLEAR all denominators by multiplying both sides by the LCM of all the denominators in the problem. 5.) The resulting equation will not have any denominators. Solve this equation. 6.) Make sure your answers do not match those numbers you found in step 2. If so, these are not solutions! 7.) Check answers into original equation to make sure they are correct. An Expression • We often simplify expressions.

An Equation • We solve equations.

VS.

Example 1. Solve each equation.

a.)

b.)

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5.5 Rational Equations Pg2 c.) d.)

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5.6 Applications and Proportions Pg 1 Key ideas: • Work Problems • Proportion Problems • Motion Problems Work Problems Example 1. Leah can wash the zoo's elephants in 3 hours. Ian, who is less experienced, needs 4 hours to do the same job. Working together, how long will it take them to wash the elephants?

Work Formula used to Solve Motion Problems:

Where is the time it takes for A to do a certain job, is the time it takes B to do the same job, and is the time it takes them to complete the job together. Example 2. Karla and William press shirts for Perfection Laundry. Each week an amusement park drops off 320 shirts to be laundered and pressed. Karla can press twice as fast as William. Together they can press 320 shirts in 11 hours. How long would it take each to press the order alone?

Proportions Example 3. The ratio of the weight of an object on the moon to the weight of an object on Earth is about 0.16 to 1. How much will a 180-lb astronaut weigh on the moon?

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5.6 Applications and Proportions Pg2 Example 4. To determine the number of trout in a lake, a conservationist catches 112 trout, tags them, and releases them back into the lake. Later, 82 trout are caught; 32 of them are tagged. How many trout are in the lake?

Motion Problems Example 5. The speed of the current in Wabash River is 3mph. Brooke's kayak can travel 4 mi upstream in the time that it takes to travel 10 mi downstream. What is the speed of the kayak in still water?

Example 6. Adventure Tours has 6 leisure-tour trolleys that travel 15 mph slower than their express-tour buses. The bus travels 132 miles in the (same) time that it takes the trolley to travel 99 miles. Find the speed of each mode of transportation.

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5.7 - Formulas and Applications Key ideas: • Know how to solve for a variable when multiple terms contain that variable. • Be able to find the LCD to clear the denominator from the problem in order to isolate a variable.

To solve for a variable when there are multiple terms that contain that variable, do the following: a.) Add or Subtract to get all the terms that contain that variable on one side, all the other terms on the other side of the equation. b.) Next, factor out the common factor (note that this common factor is the variable that you are trying to solve for. ) c.) Lastly, divide both sides to isolate the variable you are trying to solve for. Example 1. Solve the given formula for the indicated variable. a.)

b.)

solve for

solve for

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