4.5 Multiplication and Division of Rational Expressions

Transcription

1 .5. Multiplication and Division of Rational Epressions Multiplication and Division of Rational Epressions Learning Objectives Multiply rational epressions involving monomials. Multiply rational epressions involving polynomials. Multiply a rational epression by a polynomial. Divide rational epressions involving polynomials. Divide a rational epression by a polynomial. Solve real-world problems involving multiplication and division of rational epressions. Introduction The rules for multiplying and dividing rational epressions are the same as the rules for multiplying and dividing rational numbers. Lets start by reviewing multiplication and division of fractions. When we multiply two fractions we multiply the numerators and denominators separately: a b c d = a c b d When we divide two fractions we first change the operation to multiplication. Remember that division is the inverse operation of multiplication, or you can think that division is the same as multiplication by the reciprocal of the number. a b c d = a b d c The problem is completed by multiplying the numerators and denominators separately ad bc. Multiply Rational Epressions Involving Monomials Eample Multiply We follow the multiplication rule and multiply the numerators and the denominators separately = = 60 0 Notice that the answer is not in simplest form. We can divide out a common factor of 20 from the numerator and denominator of the answer. 268

2 Chapter. Rational Equations and Functions 60 0 = 3 2 We could have obtained the same answer a different way: by dividing out common factors before multiplying = We can divide out a factor of from the numerator and denominator: = We can also divide out a factor of 5 from the numerator and denominator: = = 3 2 = 3 2 Answer: The final answer is 3 2, no matter which you you go to arrive at it. Multiplying rational epressions follows the same procedure. Divide out common factors from the numerators and denominators of the fractions. Multiply the leftover factors in the numerator and denominator. Eample 2 Multiply the following a b3. 6b 8 5a 2 Divide out common factors from the numerator and denominator. a b 3 6 b 8 b 5 5 a 2 a When we multiply the left-over factors, we get Eample 3 Multiply 9 2 y2. 2 Rewrite the problem as a product of two fractions. 20ab 5 Answer 269

3 .5. Multiplication and Division of Rational Epressions y2 2 Divide out common factors from the numerator and denominator We multiply the left-over factors and get y y Answer Multiply Rational Epressions Involving Polynomials When multiplying rational epressions involving polynomials, the first step involves factoring all polynomials epressions as much as we can. We then follow the same procedure as before. Eample Multiply Factor all polynomial epressions where possible. ( + 3) 3 2 ( + 3)( 3) Divide out common factors in the numerator and denominator of the fractions: The simplifed product in factored form is: ( + 3) 3 2 ( + 3) ( 3) (Optional) Multiply the left-over factors. 3( 3) Eample 5 Multiply ( 3) = Answer

4 Chapter. Rational Equations and Functions Factor all polynomial epression where possible. (3 + 2)( 3) ( + )( ) ( + )( + 6) ( 3)( 6) Divide out common factors in the numerator and denominator of the fractions. The simplifed product in factored form is: (3 + 2) ( 3) ( + ) ( ) ( + ) ( + 6) ( 3) ( 6) (Optional) Multiply the remaining factors. (3 + 2)( + 6) ( )( 6) (3 + 2)( + 6) ( )( 6) = Answer Multiply a Rational Epression by a Polynomial When we multiply a rational epression by a whole number or a polynomial, we must remember that we can write the whole number (or polynomial) as a fraction with denominator equal to one. We then proceed the same way as in the previous eamples. Eample Multiply ( ). Rewrite the epression as a product of fractions Factor all polynomials possible and cancel common factors. The simplified product in factored form is: 3( + 6) ( + 5) ( ) ( 2) ( + 5) 3( + 6)( 2) ( ) 27

5 .5. Multiplication and Division of Rational Epressions (Optional) Multiply the remaining factors. (3 + 8)( 2) = Divide Rational Epressions Involving Polynomials Since division is the reciprocal of the multiplication operation, we first rewrite the division problem as a multiplication problem and then proceed with the multiplication as outlined in the previous eample. Note: Remember that a b c d = a b d c. The first fraction remains the same and you take the reciprical of the second fraction. Do not fall in the common trap of flipping the first fraction. Eample 7 Divide First convert the division problem into a multiplication problem by flipping what we are dividing by and then simplify as usual = = = 2 9 Eample 8 Divide First convert into a multiplication problem by flipping what we are dividing by and then simplify as usual. Factor all polynomials and divide out common factors The simplified product in factored form is: 3 ( 5) (2 + 7) ( + 3) (2 + 7) ( 2) ( 5) ( + 5) (Optional) Multiply the factors. 3( + 3) ( 2)( + 5) 3( + 3) ( 2)( + 5) =

6 Chapter. Rational Equations and Functions Divide a Rational Epression by a Polynomial When we divide a rational epression by a whole number or a polynomial, we must remember that we can write the whole number (or polynomial) as a fraction with denominator equal to one. We then proceed the same way as in the previous eamples. Eample 9 Divide (22 2 8). Rewrite the epression as a division of fractions Convert into a multiplication problem by taking the reciprocal of the divisor (i.e. what we are dividing by). Factor all polynomials and divide out common factors The simplified quotient in factored for is: (3 2) (3 + 2) 2( ) (3 2) (7 + ) 3 2 2( ) (Optional) Multiply the remaining factors Solve Real-World Problems Involving Multiplication and Division of Rational Epressions Eample 0 Suppose Marciel is training for a running race. Marciel s speed (in miles per hour) of his training run each morning is given by the function 0.( 3 9), where is the number of bowls of cereal he had for breakfast ( 6). Marciels training distance (in miles), if he eats bowls of cereal, is 0.(3 2 9). What is the function for Marciel s time and how long does it take Marciel to do his training run if he eats five bowls of cereal on Tuesday morning? 273

7 .5. Multiplication and Division of Rational Epressions If = 5, then time = distance speed time = time = = 0.3( 3) 0.( 2 9) = 0.3 ( 3) 0. ( + 3) ( 3) Answer Marciel will run for 3 8 of an hour. time = = 3 8 Review Questions Perform the indicated operation and reduce the answer to lowest terms. 3 2y2 2y y 22 y 2 3. y y y 2y y 2 y 2 9 y 3 2y 6ab 6. a a3 b 3b a a 3 a ab+b 2 (a + b) ab 2 a 2 b ( 5) (62 + 5) Marias recipe asks for 2 2 times more flour than sugar. How many cups of flour should she mi in if she uses 3 3 cups of sugar? 22. George drives from San Diego to Los Angeles. On the return trip, he increases his driving speed by 5 miles per hour. In terms of his initial speed, by what factor is the driving speed decreased on the return trip? 27

8 Chapter. Rational Equations and Functions 23. Ohm s Law states that in an electrical circuit I = V R tot. The total resistance for resistors placed in parallel is given by R tot = R + R 2. Write the formula for the electric current in term of the component resistances: R and R 2. Review Answers. 2 y y y y 3 2 2y+ y a a 9. a+b 0.. ab 2 a 2 b 2( 2)( 3) or (+)( 5) (+3)(3 5) or (+3)(5+) 6 5 or (+5) ( 3)(3 2) or (+) or cups 22. s s I = V R + V R 2 275