Rational Expressions: Multiplying and Dividing Rational Expressions

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1 OpenStax-CNX module: m2964 Rational Expressions: Multiplying and Dividing Rational Expressions Wade Ellis Denny Burzynski This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License.0 Abstract <para>this module is from <link document="col064">elementary Algebra</link> by Denny Burzynski and Wade Ellis, Jr.</para> <para>a detailed study of arithmetic operations with rational expressions is presented in this chapter, beginning with the denition of a rational expression and then proceeding immediately to a discussion of the domain. The process of reducing a rational expression and illustrations of multiplying, dividing, adding, and subtracting rational expressions are also included. Since the operations of addition and subtraction can cause the most diculty, they are given particular attention. We have tried to make the written explanation of the examples clearer by using a "freeze frame" approach, which walks the student through the operation step by step.</para> <para>the ve-step method of solving applied problems is included in this chapter to show the problem-solving approach to number problems, work problems, and geometry problems. The chapter also illustrates simplication of complex rational expressions, using the combine-divide method and the LCD-multiplydivide method.</para> <para>objectives of this module: be able to multiply and divide rational expressions.</para> Overview Multiplication Of Rational Expressions Division Of Rational Expressions 2 Multiplication Of Rational Expressions Rational expressions are multiplied together in much the same way that arithmetic fractions are multiplied together. To multiply rational numbers, we do the following: Denition : Method for Multiplying Rational Numbers. Reduce each fraction to lowest terms. 2. Multiply the numerators together.. Multiply the denominators together. Version.: May 27, :5 am

2 OpenStax-CNX module: m Rational expressions are multiplied together using exactly the same three steps. Since rational expressions tend to be longer than arithmetic fractions, we can simplify the multiplication process by adding one more step. Denition 2: Method for Multiplying Rational Expressions. Factor all numerators and denominators. 2. Reduce to lowest terms rst by dividing out all common factors. (It is perfectly legitimate to cancel the numerator of one fraction with the denominator of another.). Multiply numerators together. 4. Multiply denominators. It is often convenient, but not necessary, to leave denominators in factored form. Sample Set A Perform the following multiplications. Example Example 2 Example 4 2 = 4 2 = = )8 9 = 4 )6 9 = 4 27 Example 4 x 5y 7 2y = ) x 5y 7 y = x 7 5y 4y = )2 4 7x 20y 2 x+4 x+7 )x+4 x+7 )x+4 x+7 x+4 Divide out the common factor x + 4. Multiply numerators and denominators together. Example 5 x 2 +x 6 x 2 4x+ x2 2x x (x+)() (x )(x ) (x )(x+) (x+))() )(x )(x+) )(x )(x ) (x+6))() Factor. (x+6)() Divide out the common factors x 2 and x. Multiply. (x+)(x+) (x )(x+6) or x 2 +4x+ (x )(x+6) or x 2 +4x+ x 2 +5x 6 Each of these three forms is an acceptable form of the same answer.

3 OpenStax-CNX module: m2964 Example 6 2x+6 8x 6 x 2 4 x 2. 2(x+) 8() (x+2)() )2)(x+) (x+2))() )8)() )(x+)(x 4) 4 x+2 x+2 4(x 4) or 4x 6 Factor. (x 4)(x+) Divide out the common factors 2, x + and x 2. Multiply. Example 7 Both these forms are acceptable forms of the same answer. Example 8 x 2 x+7 x 5. x 2 x+7 x 5 x 2 (x+7) x 5 Rewrite x2 as x2. Multiply. (x ) 4x 9 x 2 6x+9. )(x ) 4x 9 )(x )(x ) 4x 9 x Example 9 x 2 x 2 +8x+5 4x+20 x 2 +2x. (x 2 +x+2) x 2 +8x+5 4x+20 x 2 +2x (x+))(x+2) 4)(x+5) (x+))(x+5) x)(x+2) 4(x+) x(x+) Factor from the rst numerator. Factor. = 4x 4x x(x+) or x 2 +x Multiply. 4 Practice Set A Perform each multiplication. Exercise (Solution on p. 0.) Exercise 2 (Solution on p. 0.) a b 2 c 2 c5 a 5 Exercise (Solution on p. 0.) y y 2 + y+ y 2 Exercise 4 (Solution on p. 0.) x 2 x 2 +7x+6 x2 4x 5 x 2 9x+20 Exercise 5 (Solution on p. 0.) x 2 +6x+8 x 2 6x+8 x2 2x 8 x 2 +2x 8

4 OpenStax-CNX module: m Division Of Rational Expressions To divide one rational expression by another, we rst invert the divisor then multiply the two expressions. Symbolically, if we let P, Q, R, and S represent polynomials, we can write P Q R S = P Q S R = P S QR 6 Sample Set B Perform the following divisions. Example 0 6x 2 5a 2x 0a. 2 2 )6x )2 ) )0a )5)a )2)x = x2a2 = 6a 2 x Invert the divisor and multiply. Example x 2 +x 0 2 x2 +9x+20 x 2 +x 4 x 2 +x 0 2 x2 +x 4 x 2 +9x+20 )(x+5)() )(x+4))(x ) 2)(x ) )(x+5))(x+4) 2 Invert and multiply. Factor. Example 2 (4x + 7) 2x+2 4x+7 2x+2 4x+7 4x+7. Write 4x + 7 as. Invert and multiply. 2x+2 Factor. )4x+7 ()4x+7) = 7 Practice Set B Perform each division. Exercise 6 (Solution on p. 0.) 8m 2 n a 5 b 2m 2 5a 7 b 2 Exercise 7 (Solution on p. 0.) x 2 4 x 2 +x 6 x2 + x 2 +4x+ Exercise 8 (Solution on p. 0.) 6a 2 +7a+2 a+2 (2a + )

5 OpenStax-CNX module: m Excercises For the following problems, perform the multiplications and divisions. Exercise 9 (Solution on p. 0.) 4a 5b b 2a Exercise 0 9x 4 4y 0y x 2 Exercise (Solution on p. 0.) a b b a Exercise 2 2x 5y 5y 2x Exercise (Solution on p. 0.) 2a a Exercise 4 9m m 2 Exercise 5 (Solution on p. 0.) 8x 6 7 4x 2 Exercise 6 4a a 5 Exercise 7 (Solution on p. 0.) 6x 6 y 5x 2 25x 4y Exercise 8 27a 7 b 4 9b a4 b 2 6a 5 Exercise 9 (Solution on p. 0.) 0x 2 y 7y 5 49y 5x 6 Exercise 20 22m n 4 m 6 n mn 4mn Exercise 2 (Solution on p. 0.) 0p 2 q 7a b 2a5 b 2 2p Exercise 22 25m 4 n 4r s 2rs4 0mn Exercise 2 (Solution on p. 0.) 9 a a 2 Exercise 24 0 b 4 2 b Exercise 25 (Solution on p. 0.) 2a 4 5b 2 4a 5b Exercise 26 42x 5 6y 4 2x4 8y Exercise 27 (Solution on p. 0.) 9x 2 y 2 55p 2 x y 5p 6 Exercise 28 4mn 25n 6 2m 20m 2 n Exercise 29 (Solution on p. 0.) 2a 2 b 5xy 4 6a2 5x 2

6 OpenStax-CNX module: m Exercise 0 24p q 9mn 0pq 2n 2 Exercise (Solution on p. 0.) x+8 x+ x+2 x+8 Exercise 2 x+0 x 4 x 4 x Exercise (Solution on p. 0.) 2x+5 x+8 x+8 Exercise 4 y+2 2y 2y y 2 Exercise 5 (Solution on p. 0.) x 5 x x 5 4 Exercise 6 x x 4 2x 5x+ Exercise 7 (Solution on p. 0.) a+2b a 4a+8b a Exercise 8 6m+2 m 4m 4 m Exercise 9 (Solution on p. 0.) x 4ab x Exercise 40 y 4 x2 y 2 Exercise 4 (Solution on p..) 2a 5 6a2 4b Exercise 42 6x 2 y 0xy Exercise 4 (Solution on p..) 2m 4 n 2 mn2 7n Exercise 44 (x + 8) x+2 x+8 Exercise 45 (Solution on p..) (x 2) x Exercise 46 (a 6) (a+2)2 a 6 Exercise 47 (Solution on p..) (b + ) 4 (b 7) b+ Exercise ( 48 b ) b (b 2 +2) 2 Exercise 49 (Solution on p..) ( x 7 ) 4 x 2 (x 7) 2 Exercise 50 (x 5) x 5 Exercise 5 (Solution on p..) (y 2) y 2 y

7 OpenStax-CNX module: m Exercise 52 (y + 6) (y+6)2 y 6 Exercise 5 (Solution on p..) (a 2b) 4 (a 2b)2 a+b Exercise 54 x 2 +x+2 x 2 4x+ x2 2x 2x+2 Exercise 55 (Solution on p..) 6x 42 x 2 2x x2 x 7 Exercise 56 a+b a 2 4a 5 9a+9b a 2 a 0 Exercise 57 (Solution on p..) a 2 4a 2 a 2 9 Exercise 58 b 2 5b+6 b 2 b 2 a2 5a 6 a 2 +6a+9 b2 2b b 2 9b+20 Exercise 59 (Solution on p..) m 2 4m+ m 2 +5m 6 m2 +4m 2 m 2 5m+6 Exercise 60 r 2 +7r+0 r 2 2r 8 r2 +6r+5 r 2 r 4 Exercise 6 (Solution on p..) 2a 2 +7a+ a 2 5a 2 a2 5a+6 a 2 +2a Exercise 62 6x 2 + 2x 2 +7x 4 x2 +2 x 2 4x 4 Exercise 6 (Solution on p..) x y x 2 y 2 x 2 y y 2 x2 y x xy Exercise 64 4a b 4a 2 b 2 a 2 5a 0 4ab 2b 2 Exercise 65 (Solution on p..) x+ x 4 x 4 x+ x+ Exercise 66 x 7 x+8 x+ x 7 x+8 Exercise 67 (Solution on p..) 2a b a+b a+b a 5b a 5b 2a b Exercise 68 a(a+) 2 a 5 6(a 5)2 5a+5 5a+0 4a 20 Exercise 69 (Solution on p..) a 2 4b 8b 5a Exercise 70 6x 5y 2 20y 2x Exercise 7 (Solution on p..) 8x 2 y 5x 4 5xy Exercise 72 4a b 2a 6b 2 Exercise 7 (Solution on p..) a 2a+2 a2 a+2 a 2 5a 6

8 OpenStax-CNX module: m Exercise 74 x 2 x 2 x 4 x2 +2x+ 4x 8 Exercise 75 (Solution on p..) 5x 0 x 2 4x+ x2 +4x+ x 2 + Exercise 76 a 2 2a+5 6a 2 a2 2a 8 2a 0 Exercise 77 (Solution on p..) b 2 5b+4 b 6 Exercise 78 a+6 4a 24 6 a a+5 b2 9b 4 b+8 Exercise 79 (Solution on p..) 4x+2 x 7 7 x 2x+2 Exercise 80 2b 2 b 2 +b 6 b+2 b+5 Exercise 8 (Solution on p..) x 2 6x 9 2x 2 6x 4 x2 5 6x 2 7x Exercise 82 2b 2 2b+4 8b 2 28b 6 b2 2b+ 2b 2 5b Exercise 8 (Solution on p..) x 2 +4x+ x 2 +5x+4 (x + ) Exercise 84 x 2 x+2 x 2 4x+ (x ) Exercise 85 (Solution on p..) x 2 2x+8 x 2 +5x+6 (x + 2) 9 Exercises For Review Exercise 86 ( here ) If a < 0, then a =. Exercise 87 (Solution on p. 2.) ( here 2 ) Classify the polynomial 4xy + 2y as a monomial, binomial, or trinomial. State its degree and write the numerical coecient of each term. Exercise 88 ( here ) Find the product: y 2 (2y ) (2y + ). Exercise 89 (Solution on p. 2.) ( here 4 ) Translate the sentence four less than twice some number is two more than the number into an equation. "Basic Operations with Real Numbers: Absolute Value" < 2 "Algebraic Expressions and Equations: Classication of Expressions and Equations" < "Algebraic Expressions and Equations: Combining Polynomials Using Addition and Subtraction" < 4 "Solving Linear Equations and Inequalities: Application I - Translating from Verbal to Mathetical Expressions" <

9 OpenStax-CNX module: m Exercise 90 ( here 5 ) Reduce the fraction x2 4x+4 x "Rational Expressions: Reducing Rational Expressions" <

10 OpenStax-CNX module: m Solutions to Exercises in this Module Solution to Exercise (p. ) 0 7 Solution to Exercise (p. ) c a 2 b 2 Solution to Exercise (p. ) y 2 + Solution to Exercise (p. ) x+ x+6 Solution to Exercise (p. ) (x+2) 2 () 2 Solution to Exercise (p. 4) 20a 2 mn Solution to Exercise (p. 4) x+ x Solution to Exercise (p. 4) a+4 a+2 6a 2 5 6a 2 5 9x x 5 y 2 4 x 4 y 5a 2 bpq a 9a b 2 9p 4 y x 6b x y 4 x+2 x+ 2x+5 4 x 4

11 OpenStax-CNX module: m2964 4abx 2 4a b 49m n x (b + ) (b 7) ( x 7 ) 2 (x + ) (x ) (y ) (a 2b) 2 (a + b) 6(x ) (x ) (a+2)(a+) (a )(a+) (2a+)(a 6)(a+) (a+)(a )(a 2) x(x y) y x+ a+b a+b 2ab 2 5 6x 2 y 4 (a 2)(a ) 2(a 6)(a+) 5(x 2 +4x+) (x )(x ) 2 (b 8) (b+2) 2(x+) (x+) (x )(x+)(2x ) 2(x 2 )() (x+4)(x ) (x+)(x 2 4x )

12 OpenStax-CNX module: m (x 6)(x ) (x+2) 2 (x+) binomial; 2; 4, 2 2x 4 = x + 2