Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Chapter 12 Rational Expressions

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Simplifying Rational Expressions

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 33 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Rational Expression A rational expression is an expression that can be written in the form where P and Q are both polynomials and Q ≠ 0. Examples of Rational Expressions

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 44 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. To evaluate a rational expression for a particular value(s), substitute the replacement value(s) into the rational expression and simplify the result. Evaluating Rational Expressions Example Evaluate the following expression for y = 2.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 55 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. In the previous example, what would happen if we tried to evaluate the rational expression for y = 5? Since division by 0 is undefined, this expression is undefined when y = 5. Evaluating Rational Expressions

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 66 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. To find values for which a rational expression is undefined, find values for which the denominator is 0. Undefined Rational Expressions

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 77 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Find any values that make the following rational expression undefined. The expression is undefined when 15x + 45 = 0. So, the expression is undefined when x = 3. Undefined Rational Expressions Example

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 88 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Simplifying a rational expression means writing it in lowest terms or simplest form. To do this, we need to use the Fundamental Principle of Rational Expressions If P, Q, and R are polynomials, and Q and R are not 0, Simplifying Rational Expressions

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 99 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. To Simplify a Rational Expression Step 1:Completely factor the numerator and denominator. Step 2:Divide out factors common to the numerator and denominator. (This is the same as “removing a factor of 1.”) Simplifying Rational Expressions

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 10 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Helpful Hint When simplifying a rational expression, we look for common factors, not common terms.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 11 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. x 2 + 5x Simplifying Rational Expressions Example 7x + 35 Simplify:

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 12 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Simplify: Simplifying Rational Expressions Example x 2 – x – 20 x 2 + 3x – 4

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 13 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Simplify: Simplifying Rational Expressions Example y – 7 7 – y

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Multiplying and Dividing Rational Expressions

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 15 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Multiplying Rational Expressions Multiplying rational expressions If are rational expressions, then To multiply rational expressions, multiply the numerators and then multiply the denominators.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 16 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. After multiplying such expressions, our result may not be in simplified form, so we use the following techniques. To Multiply Rational Expressions Step 1:Completely factor numerators and denominators. Step 2:Multiply numerators and multiply denominators. Step 3:Simplify or write the product in lowest terms by dividing out common factors. Multiplying Rational Expressions

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 17 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Multiply: Example Multiplying Rational Expressions 10x 3 6x26x2 12 5x5x

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 18 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Multiply: Multiplying Rational Expressions Example m + n (m – n) 2 m m 2 – mn

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 19 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Dividing rational expressions If are rational expressions, then Dividing Rational Expressions To divide two rational expressions, multiply the first rational expression by the reciprocal of the second rational expression.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 20 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Remember, to Divide by a Rational Expression, multiply by its reciprocal. Helpful Hint

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 21 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Divide: Dividing Rational Expressions Example

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 22 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Converting Between Units of Measure Use unit fractions (equivalent to 1), but with different measurements in the numerator and denominator. Multiply the unit fractions like rational expressions, canceling common units in the numerators and denominators. Units of Measure

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 23 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall square inches = _________ square feet. (1008 sq in) (2·2·2·2·3·3·7 in · in) Example Units of Measure

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Adding and Subtracting Rational Expressions with the Same Denominator and Least Common Denominators

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 25 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Rational Expressions Adding and Subtracting Rational Expressions with Common Denominators If are rational expressions, then and To add or subtract rational expressions, add or subtract numerators and place the sum or difference over the common denominator.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 26 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Add. Adding Rational Expressions Example

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 27 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Subtract: Subtracting Rational Expressions Example

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 28 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Subtract: Subtracting Rational Expressions Example

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 29 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. To add or subtract rational expressions with different denominators, you have to change them to equivalent forms that have the same denominator (a common denominator). This involves finding the least common denominator of the two original rational expressions. Least Common Denominators

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 30 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. To Find the Least Common Denominator (LCD) Step 1: Factor each denominator completely. Step 2:The least common denominator (LCD) is the product of all unique factors found in Step 1, each raised to a power equal to the greatest number of times that the factor appears in any one factored denominator. Least Common Denominators

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 31 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Find the LCD of the following rational expressions. Least Common Denominators Example

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 32 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Find the LCD of the following rational expressions. Least Common Denominators Example

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 33 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Find the LCD of the following rational expressions. Least Common Denominators Example

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 34 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Find the LCD of the following rational expressions. Both of the denominators are already factored. Since each is the opposite of the other, you can use either x – 3 or 3 – x as the LCD. Least Common Denominators Example

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 35 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. To change rational expressions into equivalent forms, we use the principal that multiplying by 1 (or any form of 1), will give you an equivalent expression. Writing Equivalent Rational Expressions

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 36 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Rewrite the rational expression as an equivalent rational expression with the given denominator. Equivalent Expressions Example

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Adding and Subtracting Rational Expressions with Different Denominators

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 38 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. As stated in the previous section, to add or subtract rational expressions with different denominators, we have to change them to equivalent forms first. Different Denominators

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 39 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. To Add or Subtract Rational Expressions with Different Denominators Step 1: Find the LCD of all the rational expressions. Step 2: Rewrite each rational expression as an equivalent expression whose denominator is the LCD found in Step 1. Step 3: Add or subtract numerators and write the sum or difference over the common denominator. Step 4: Simplify or write the rational expression in lowest terms. Different Denominators

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 40 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Add the following rational expressions. Adding Rational Expressions with Different Denominators Example

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 41 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Subtract: Example Subtracting Rational Expressions With Different Denominators

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 42 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Subtract: Subtracting Rational Expressions With Different Denominators Example

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 43 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Add: Example Adding Rational Expressions with Different Denominators

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Solving Equations Containing Rational Expressions

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 45 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Solving Equations Containing Radical Expressions To solve equations containing rational expressions, clear the fractions by multiplying both sides of the equation by the LCD of all the fractions. Then solve the simplified equation as in previous sections.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 46 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Solve: Example Multiply both sides by the LCD, 6x. Apply the distributive property. Simplify. Continued Solving Equations Containing Radical Expressions

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 47 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. True To check, we replace x with 10 in the original equation. Check: Solving Equations Containing Radical Expressions Example continued

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 48 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. To Solve an Equation Containing Rational Expressions Step 1:Multiply both sides of the equation by the LCD of all rational expressions in the equation. Step 2:Remove any grouping symbols and solve the resulting equation. Step 3:Check the solution in the original equation. Solving Equations Containing Radical Expressions

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 49 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Solve the following rational equation. Example Continued Solving Equations Containing Radical Expressions Multiply by the LCD. Use the distributive property (twice) Simplify.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 50 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. True Check: Substitute the value for x into the original equation. The solution is Solving Equations Containing Radical Expressions Example continued

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 51 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Solve the following rational equation. Example Continued Solving Equations Containing Radical Expressions

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 52 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Check: Substitute the value for x into the original equation. True The solution is x = 3. Solving Equations Containing Radical Expressions Example continued

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 53 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Solve the following rational equation. Example Continued Solving Equations Containing Radical Expressions

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 54 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Check: Substitute the value for x into the original equation. Notice that 3 makes a denominator 0 in the original equation. Therefore, 3 is not a solution and the equation has no solution. Solving Equations Containing Radical Expressions Example continued

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 55 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Helpful Hint As we can see from the previous example, it is important to check the proposed solution(s) in the original equation.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 56 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Solve the following rational equation. Example Remember to check your answer. Solving Equations Containing Radical Expressions

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 57 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Solve the following equation for R 1. Example Solving Equations for a Specified Variable

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Rational Equations and Problem Solving

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 59 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Rational Equations and Problem Solving In this section, we will solve problems that can be modeled by equations containing rational expressions. To solve these problems, we use the same problem-solving steps that we have learned earlier.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 60 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Finding an Unknown Number Example Continued The quotient of a number and 9 times its reciprocal is 1. Find the number. Read and reread the problem. If we let n = the number, then = the reciprocal of the number 1. UNDERSTAND

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 61 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Continued Finding an Unknown Number 2. TRANSLATE Example continued The quotient of ÷ a number n and 9 times its reciprocalis = 1 1

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 62 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 3. SOLVE Continued Finding an Unknown Number Example continued

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 63 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 4. INTERPRET Finding an Unknown Number Check: We substitute the values we found from the equation back into the problem. Note that nothing in the problem indicates that we are restricted to positive values. State: The missing number is 3 or –3. True Example continued

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 64 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Solving Problems About Work Example Continued An experienced roofer can roof a house in 26 hours. A beginner needs 39 hours to do the same job. How long will it take if the two roofers work together? Read and reread the problem. By using the times for each roofer to complete the job alone, we can figure out their corresponding work rates in portion of the job done per hour. 1. UNDERSTAND Experienced roofer 26 1/26 Beginner roofer 39 1/39 Together t 1/t Time in hrsPortion job/hr

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 65 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Continued Solving Problems About Work 2. TRANSLATE Since the rate of the two roofers working together would be equal to the sum of the rates of the two roofers working independently, Example continued

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 66 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 3. SOLVE Continued Solving Problems About Work Example continued

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 67 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 4. INTERPRET Solving Problems About Work Check: We substitute the value we found from the proportion calculation back into the problem. State: The roofers would take 15.6 hours working together to finish the job. True Example continued

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 68 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Continued Solving Problems About Distance These problems involve the distance formula, d = r t.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 69 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Finding a Speed Example Continued The speed of Lazy River’s current is 5 mph. A boat travels 20 miles downstream in the same time as traveling 10 miles upstream. Find the speed of the boat in still water. Read and reread the problem. By using the formula d=rt, we can rewrite the formula to find that t = d/r. We note that the rate of the boat downstream would be the rate in still water + the water current and the rate of the boat upstream would be the rate in still water – the water current. 1. UNDERSTAND Down 20 r /(r + 5) Up 10 r – 5 10/(r – 5) Distance rate time = d/r

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 70 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Continued Finding a Speed 2. TRANSLATE Since the problem states that the time to travel downstairs was the same as the time to travel upstairs, we get the equation Example continued

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 71 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 3. SOLVE Continued Finding a Speed Example continued

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 72 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 4. INTERPRET Finding a Speed Check: We substitute the value we found from the proportion calculation back into the problem. True State: The speed of the boat in still water is 15 mph. Example continued

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Simplifying Complex Fractions

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 74 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Complex Rational Fractions Complex rational expressions or complex fractions are rational expressions whose numerator, denominator, or both contain one or more rational expressions. There are two methods that can be used when simplifying complex fractions.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 75 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Method 1: To Simplify a Complex Fraction Step 1:Add or subtract fractions in the numerator or denominator so that the numerator is a single fraction and the denominator is a single fraction. Step 2:Perform the indicated division by multiplying the numerator of the complex fraction by the reciprocal of the denominator of the complex fraction. Step 3:Write the rational expression in lowest terms. Simplifying Complex Fractions— Method 1

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 76 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Example Simplify the complex fraction Simplifying Complex Fractions

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 77 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Method 2: To Simplify a Complex Fraction Step 1:Find the LCD of all the fractions in the complex fraction. Step 2:Multiply both the numerator and the denominator of the complex fraction by the LCD from Step 1. Step 3: Perform the indicated operations and write the result in lowest terms. Simplifying Complex Fractions — Method 2

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 78 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Simplifying Complex Fractions Example Simplify the complex fraction