Multiplying and Dividing Algebraic Fractions


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1 . Multiplying and Dividing Algebraic Fractions. OBJECTIVES. Write the product of two algebraic fractions in simplest form. Write the quotient of two algebraic fractions in simplest form. Simplify a comple fraction by the method of common denominators In arithmetic, you found the product of two fractions by multiplying the numerators and the denominators. For eample, 7 7 In symbols, we have Rules and Properties: Multiplying Algebraic Fractions NOTE P, Q, R, and S again represent polynomials. P Q R S PR QS when Q 0 and S 0 It is easier to divide the numerator and denominator by any common factors before multiplying. Consider the following. NOTE Divide by the common factors of and. The alternative is to multiply first: 9 7 and then use the GCF to reduce to lowest terms Step by Step: Step Step Step In algebra, we multiply fractions in eactly the same way. To Multiply Algebraic Fractions Factor the numerators and denominators. Divide the numerator and denominator by any common factors. Write the product of the remaining factors in the numerator over the product of the remaining factors in the denominator. 00 McGrawHill Companies NOTE Divide by the common factors of,, and y. Eample illustrates this property. Eample Multiplying Algebraic Fractions Multiply the following fractions. (a) y 0y 0y y y
2 CHAPTER ALGEBRAIC FRACTIONS NOTE Divide by the common factors of,, and. (b) 9 ( ) ( ) 9 ( ) ( ) 9 Factor NOTE ( ) (c) 0 ( ) ( ) ( ) ( ) NOTE Divide by the common factors of,, and. (d) (e) y y 0y y y ( )( ) ( ) ( y) y y ( y) ( y) ( ) 0 y ( y) ( y) CHECK YOURSELF Multiply. (a) (b) y 0y (c) (d) (e) 9 NOTE Recall, of. is the reciprocal NOTE Once more P, Q, R, and S are polynomials. You can also use your eperience from arithmetic in dividing fractions. Recall that, to divide fractions, we invert the divisor (the second fraction) and multiply. For eample, In symbols, we have Rules and Properties: P Q R S P Q S R PS QR when Q 0, R 0, and S 0. Dividing Algebraic Fractions Division of algebraic fractions is done in eactly the same way. 00 McGrawHill Companies
3 MULTIPLYING AND DIVIDING ALGEBRAIC FRACTIONS SECTION. Step by Step: To Divide Algebraic Fractions Step Step Invert the divisor and change the operation to multiplication. Proceed, using the steps for multiplying algebraic fractions. Eample illustrates this approach. Dividing Algebraic Fractions Divide the following. (a) (b) Eample 9 y y 9 y 9 9 y y y 9 Invert the divisor and multiply. No simplification can be done until the divisor is inverted. Then divide by the common factors of and. NOTE Factor all numerators and denominators before dividing out any common factors. (c) y y y 9 y y ( y) 9 y y 9 y y ( ( y) ( y) y) (d) ( ) ( ) ( ) ( ) ( ) 00 McGrawHill Companies CHECK YOURSELF Divide. (a) (b) y 7 y 0y 9y (c) (d) 9 0y y 0y 0
4 CHAPTER ALGEBRAIC FRACTIONS Before we continue, let s review why the invertandmultiply rule works for dividing fractions. We will use an eample from arithmetic for the eplanation. Suppose that we want to divide as follows: () We can write () Interpret the division as a fraction. We are multiplying by. () We then have Comparing epressions () and (), you should see the rule for dividing fractions. Invert the fraction that follows the division symbol and multiply. A fraction that has a fraction in its numerator, in its denominator, or in both is called a comple fraction. For eample, the following are comple fractions NOTE This is the Fundamental Principle of Fractions. and a a Remember that we can always multiply the numerator and the denominator of a fraction by the same nonzero term. P in which Q 0 and R 0 Q P R Q R To simplify a comple fraction, multiply the numerator and denominator by the LCD of all fractions that appear within the comple fraction. Eample Simplifying Comple Fractions Simplify. 00 McGrawHill Companies
5 MULTIPLYING AND DIVIDING ALGEBRAIC FRACTIONS SECTION. 7 The LCD of and is. So multiply the numerator and denominator by. CHECK YOURSELF Simplify. (a) 7 7 (b) The same method can be used to simplify a comple fraction when variables are involved in the epression. Consider Eample. Eample Simplifying Comple Algebraic Fractions NOTE Be sure to write the result in simplest form. Simplify. 0 0 The LCD of and is, so multiply the numerator and denominator by CHECK YOURSELF Simplify. 00 McGrawHill Companies (a) 9 We may also have a sum or a difference in the numerator or denominator of a comple fraction. The simplification steps are eactly the same. Consider Eample. (b) m m 0
6 CHAPTER ALGEBRAIC FRACTIONS Eample Simplifying Comple Algebraic Fractions y Simplify. y The LCD of,,, and is y, so multiply the numerator and denominator by y. y y NOTE Notice the use of the distributive property to multiply each term in the numerator and in the denominator by y. y y y y y y y y y y y y CHECK YOURSELF Simplify. y y y y NOTE To divide by a fraction, we invert the divisor (it follows the division sign) and multiply. P Q R S A second method for simplifying comple fractions uses the fact that P Q R S P Q S R To use this method, we must write the numerator and denominator of the comple fraction as single fractions. We can then divide the numerator by the denominator as before. The following algorithm summarizes our work with simplifying comple fractions. Step by Step: To Simplify Comple Fractions Step Step Multiply the numerator and denominator of the comple fraction by the LCD of all the fractions that appear within the comple fraction. Write the resulting fraction in simplest form. CHECK YOURSELF ANSWERS y. (a) ; (b) 0; (c) ; (d) ; (e) ( ). (a) ; (b) ; (c) ; (d) y 9 m. (a) ; (b). (a) ; (b). 0 ( ) ( 7) y y 00 McGrawHill Companies
7 Name. Eercises Section Date Multiply ANSWERS... y w.. a.. a y 7.. 0y y 9y y y y y 7. ab ab a b 7y y y y m n.. 0mn mn 9mn McGrawHill Companies p p p.. p p m m 7.. m m 7m m 9 r r 9r r r 0 9r a a 9a a a 7a a ab a ab b a 9b a ab
8 ANSWERS. y.. y y 7 y 0y a 9b a ab b a ab 7a b Divide w w 9.. y y 7y y 7y y y a.. a a a a p 9p p 9 p p a 9b a ab a ab b ab m m 7 m 9 r rs s r r s m m m m r 9s r 00 McGrawHill Companies 0
9 ANSWERS y.. y ( y) p pq q p q (p pq) Perform the indicated operations m n m mn m m mn n Simplify each comple fraction. m n m mn m 0 m McGrawHill Companies. a. a y 7. y y. y 9 w w w w
10 ANSWERS a a y.. y y.. y.. y y a b a b r r r r y 7.. y y y y y y y y a a y Ecology. Herbicides constitute of all pesticides used in the United States. Insecticides are of all pesticides used in the United States. The ratio of herbicides to insecticides used in the United States can be written. Write this ratio in simplest form. 7. Ecology. Fungicides account for of the pesticides used in the United States. 0 Insecticides account for of all the pesticides used in the United States. The ratio of fungicides to insecticides used in the United States can be written. Write this 0 ratio in simplest form. 00 McGrawHill Companies
11 ANSWERS 77. Ecology. The ratio of insecticides to herbicides applied to wheat, soybeans, corn, and 7 cotton can be epressed as. Simplify this ratio Find the area of the rectangle shown Find the area of the rectangle shown. 0. Electricity. The combined resistance of two resistors R and R in parallel is given by the formula R T R R Simplify the formula.. Comple fractions have some interesting patterns. Work with a partner to evaluate each comple fraction in the sequence below. This is an interesting sequence of fractions because the numerators and denominators are a famous sequence of whole numbers, and the fractions get closer and closer to a number called the golden mean. 00 McGrawHill Companies,,,,,,,,,,,, After you have evaluated these first five, you no doubt will see a pattern in the resulting fractions that allows you to go on indefinitely without having to evaluate more comple fractions. Write each of these fractions as decimals. Write your observations about the sequence of fractions and about the sequence of decimal fractions.
12 ANSWERS a. b. c. d. e. f. Solve each of the following equations. (a) 0 (b) (c) (d) ( ) (e) ( ) (f) ( ) ( ) Answers Getting Ready for Section. [Section.]. y a y a n. p m r p m r 7. y 9 a a 9. b a b ( ) 9. a (y ) y y. y y y y (y )(y ) a. b. m c. d. e. f. 00 McGrawHill Companies
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