Chapter 5 Rational Expressions

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1 Worksheet 4 (5.1 Chapter 5 Rational Expressions 5.1 Simplifying Rational Expressions Summary 1: Definitions and General Properties of Rational Numbers and Rational Expressions A rational number can be written as a quotient of two integers, in the form where the denominator, b, is not 0. A rational expression is the indicated quotient of two polynomials where value of the denominator is assumed to be nonzero. a, b the Sign rules of rational numbers and expressions: a a a 1. b b b a a. b b Fundamental Principle of Fractions: a k a If b and k are nonzero integers and a is any integer, then. b k b Simplifying a rational expression: 1. Completely factor the polynomial given in the numerator and. Apply the fundamental principle of fractions by dividing the common 3. The simplest form will be the quotient of the product of remaining 97

2 Worksheet 4 (5.1 Warmup 1. Reduce to lowest terms: a yz b 0y 3 y z 5 y m 18 (m3( c m 6m9 ( ( 98

3 d x 4 ( 10 5x ( 5( x (1 or x 5 Problems Simplify: 3 90 c 1. a 96 abc 9 y 7y. 3 y 7 3. ay 3a 4y 1 ay 4a 8y 16 a a Worksheet 5 (5. 5. Multiplying and Dividing Rational Expressions Summary 1: Multiplying Rational Expressions Basic definition for multiplying rational numbers: If a, b, c, and d are integers with b and d not equal to zero, a c a c ac then. b d b d bd Multiplying rational expressions: 1. Completely factor each numerator and denominator.. Apply the basic definition for multiplying rational numbers by rewriting the numerator as a product of factors and rewrite the denominator as a product of factors. 3. Simplify by dividing common factors. 4. The result is the quotient of the product of remaining factors in the numerator and the product of remaining factors in the denominator. 99

4 Warmup 1. Multiply and simplify: a b 5ab a 7 a 49 5ab( ( a 7a a(a7( c a a 15 3a 8a 3 (a5( ( a 9 6 a 17a 5 (a 3(a 3( Worksheet 5 (5. ( ( Problems Perform the indicated operation. Express in simplest form: xy 0x 50. x x5 xy 5y 4 n 4n 4 3n 8n n 5n 9 n Summary : Dividing Rational Expressions 100

5 Basic definition for dividing rational numbers: If a, b, c, and d are integers with b, c, and d not equal to zero, then a c a d ad. b d b c bc c d Note: and are called reciprocals or multiplicative inverses. d c Dividing rational expressions: 1. Apply the basic definition for dividing rational numbers.. Follow the steps for multiplying rational expressions in summary 1. Worksheet 5 (5. Warmup. Divide and simplify: a x b 5 y 4 x y 4 x 15xy 5 y m m m m 3m m 1 c 3m 14m 5 m 7m 9 3m 14m 5 (3m 1(m1( ( ( ( (3 m(9 m Problems Perform the indicated operation. Express in simplest form:

6 5. 5xy y 7 y 7y y xy xb cy cb xy xb cy cb Worksheet 6 ( x 45 x 7x 3 3 x 7x 5.3 Adding and Subtracting Rational Expressions Adding Rational Expressions Basic definition for adding rational numbers with a common denominator: a c a c If a, b, and c are integers and b is not zero, then. b b b Equivalent fractions are fractions with different denominators that name the same number. The LCD, least common denominator, is the least common multiple of a set of denominators. The LCD can be determined by inspection or by writing the product of the highest power for each factor in any given polynomial. Adding rational expressions: 1. Determine if the given rational expressions have a common denominator.. If not, find the LCD. 3. Rewrite the given rational expressions as equivalent rational expressions using the LCD in the denominators. 4. Apply the basic definition for adding rational numbers with a common denominator. 5. Express result in simplest form. Summary 1: Warmup 1. Add and simplify: a x x 4 x x x 10

7 ( x Worksheet 6 ( b c x x 5 5 (4 1(4 7 (3 16( The LCD is. 3( x ( x ( 5 ( d 3 x 1 x 5 10 x 10 x 10 x The LCD is. Problems Add and express in simplest form: Worksheet 6 ( n6 n ( ( (x1( (x 5( 3x (x 5(x1 (x 5(x1 7x (x 5(x1 103

8 x 1 4x 1 x1 14. x 9 x3 Subtracting Rational Expressions Basic definition for subtracting rational numbers with a common denominator: a c a c If a, b, and c are integers and b is not zero, then. b b b Subtracting rational expressions: Summary : 1. Determine if the given rational expressions have a common denominator.. If not, find the LCD. 3. Rewrite the given rational expressions as equivalent rational expressions using the LCD in the denominator. 4. Apply the basic definition for subtracting rational numbers with a common denominator. 5. Express in simplest form. Warmup. Subtract and simplify: Worksheet 6 (

9 a 4m 0 m 5 m 5 ( ( m 5 4( ( b x 5 x3 x (x( (x3( (5( (x( ( x(x 3 ( ( The LCD is. x(x 3 x(x 3 ( x(x 3 5x (5x( c x 3 (x3( (( (1( The LCD is. (x3 (x3 5x ( (x3 5x x3 Worksheet 6 (5.3 Problems Subtract and simplify: 5. x1 4 x 6 105

10 x x 7 8x 7. 5 x 1 3 x3 Worksheet 7 ( More on Rational Expressions and Complex Fractions Summary 1: Complex fractions are fractional forms that contain rational numbers or rational expressions in the numerators and/or denominators. 106

11 Simplifying a complex fraction Method A: 1. If necessary, perform the indicated operation in the numerator and/or denominator.. Rewrite the indicated quotient as the division of two rational numbers or rational expressions. 3. Apply the basic definition for dividing rational numbers. Simplifying a complex fraction Method B: 1. Find the LCD of the rational numbers or rational expressions in the given complex fraction.. Multiply both numerator and denominator by the LCD to obtain an equivalent fraction. (If necessary, apply the distributive property. 3. Simplify the resulting expression. Warmup 1. Simplify using method A: 3 a 5 ( ( ( 1 ( 19 ( 1 ( Worksheet 7 (5.4 3 ( 3 b x x x 5 ( 5 1 x x x ( x ( x x 3 ( x ( x 3 ( x ( 107

12 Simplify using method B: 5 5 ( xy y c xy y ( x y x y 5 ( xy 6 ( ( x 5y ( 6 y ( y 3 y ( 6 ( d n ( n 3 6 n3 1 5 n 3 ( ( (6 ( (5( n 3 1 n3 Worksheet 7 (5.4 6n( 5n( 1 Problems Simplify: x 8 5xy 108

13 . 8 6 y y 4 x x 3. 5 x x 7 5 x 4 x Worksheet 8 ( Dividing Polynomials Summary 1: Dividing a polynomial by a monomial: 1. Divide each term of the polynomial by the common monomial divisor.. Reduce each fraction to its simplest form. Dividing a polynomial by a polynomial: 1. Use the conventional long division format and arrange both the dividend and the divisor in descending order.. The first term of the quotient is determined by dividing the first term in the dividend by the first term of the divisor. 3. Multiply each term of the divisor by the term of the quotient found in step. Position this product so that it can be subtracted from the dividend. 4. Subtract. 5. Repeat the process beginning with step using the resulting polynomial from step 4 as the new dividend. 109

14 Warmup 1. Divide: 6 a a b 3 b x a b1ab 3a x 3x 3 x1 a 3a 3a ( ( 3 3 ( a 3 The quotient is. x 1 x 3 x 3x 3 ( 3x 3 ( 0 The remainder is. Worksheet 8 (5.5 x 4 c ( x 3x1 ( x 5 x 10 x 5 x 4 0x 3 0x 3x 1 ( 10x 3x 1 ( 3x 51 ( x 10 x 5 Problems Divide: a b 36 ab 7 a b 9ab. x 4 1 x 1 110

15 3 3. (5 y y y 1 ( y Worksheet 9 ( Fractional Equations Summary 1: The restricted values of a given rational expression are those values that Solving make equations the denominator involving equal rational to 0. expressions: 1. Determine the restricted values for each rational expression in the given equation.. Find the LCD in the equation. 3. Apply the multiplication property of equality, using the LCD, to clear the equation of all fractional forms. 4. Solve the resulting equivalent equation. 5. Check when directed to do so. Note: If a restricted value results as a solution to the equation found in step 4, then the solution set is. Warmup 1. Solve: a x 4x The restricted value is, (x. The LCD is. ( ( 6 3x 4x ( ( ( 6 3x 4x 10x ( ( 10x x 111

16 The solution set is { }. Worksheet 9 (5.6 b y 5 3y 5y 10 The restricted value is, (y. The LCD is. ( 3y ( y 5 5(y ( 3y ( ( y 5 5(y 10 ( ( 3y 6 ( 3y y The solution set is. Problems Solve: x 1 x 4 x x. a 3 a 1 6 a 1 Worksheet 9 (5.6 11

17 Summary : Ratio and Proportion A ratio is the comparison of two numbers with division. A proportion is a statement of equality between two ratios. The CrossMultiplication Property of Proportions: If a b where b0 and d0, then ad bc. c d Solving fractional equations as proportions: 1. Find restricted values for each rational expression in the given equation.. Determine if the given equation is in the form a c b d. 3. If so, apply the crossmultiplication property of proportions. 4. Solve resulting equivalent equation. 5. Check when directed to do so. Warmup. Solve: a Problem Solve: x x 1 10( 6( 0x 10 ( ( x x The solution set is { } x 1 4 x 1 Worksheet 9 (5.6 Summary 3: Word Problems Using Fractional Equations Follow the listed steps in Summary 3 for Section.4. Warmup 3. Set up and write an algebraic equation, then solve: a The sum of two numbers is 40. If the larger is divided by the smaller, the quotient is 5 and the remainder is 4. Find the numbers. 113

18 Declare the variable: Let x larger number smaller number Write an algebraic equation and solve: x ( 5 ( 40 x x 4 ( ( 5 40 x 40 x x 4 ( ( ( 5 ( 40 x 40 x x 00 5x( 6x ( x The numbers are and. Problem Set up and write an algebraic equation, then solve: 4. The perimeter of a rectangle is 104 meters. If the ratio of its width to its length is 5 to 8, find the dimensions of the rectangle. Worksheet 30 ( More Fractional Equations and Applications Summary 1: The restricted values of a given rational expression are those values that Solving make fractional the denominator equations: equal to Factor the denominators.. Determine the restricted values for each rational expression in the given equation. 3. Find the LCD in the equation. 4. Apply the multiplication property of equality, using the LCD, to clear the equation of all fractional forms. 5. Solve the resulting equivalent equation. 6. Check when directed to do so. Note: If a restricted value results as a solution to the equation found in step 5, then the solution set is. 114

19 3x 1 5x 1 Warmup 1. Solve: a x 9 x3 3x 1 5x 1 x3 ( ( x3 x 3 x3 x 3 The restricted values are and, (x and x. The LCD is. 3x 1 5x 1 ( (x3(x 3 x3 x3 ( x 3 3x 1 (x3(x ( 3 5x 1 ( x3 x3 ( x 3 (3x 1 ( (5x 1 ( (x 3 ( Worksheet 30 (5.7 3x 1 5x 16x 3 ( ( 0 ( ( 0 x and x The solution set is { } Problems Solve: x x x4 x 16 x 4. 4 a a 1 3 a a a a a 3 115

20 Worksheet 30 (5.7 Word Problems Using Fractional Equations Follow the listed steps in Summary 3 for Section.4. Summary : Warmup. Set up and write an algebraic equation, then solve: a A tank can be filled by the hotwater faucet in 4 minutes and by the coldwater faucet in 6 minutes. How long does it take to fill the tank using both faucets together? Declare the variable: Let x time required using both faucets Write an algebraic equation and solve: 116

21 ( 1 4 ( ( ( ( 1 1 ( 6 x 3x ( ( 5x ( x The amount of time it takes to fill the tank with both faucets is minutes. Problems Set up and write an algebraic equation, then solve: Worksheet 30 ( John can pick a bushel of peaches in 30 minutes. His little sister can pick a bushel of peaches in 45 minutes. How long will it take them to pick a bushel of peaches working together? 4. One pipe can fill a tank in 6 hours. The drain can empty the tank in 10 hours. How long will it take to fill the tank if the drain is left open? 117

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