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If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. Regular expressions are multiplied and divided in the same way as number fractions. To multiply, first find the most significant common factors of the numerator and denominator. Then group the factors so that the fractions are equal to one. Then multiply the remaining factors. To divide, first rewrite the division as multiplication by the inverse of the denominator. The steps are the same as for multiplication. Benefits of Multiplying and Dividing Rational Expressions WorksheetsCuemath's interactive math worksheets consist of visual simulations to help your child visualize the concepts being taught, i.e., "see things in action and reinforce learning from it." The Multiplying and dividing rational expressions worksheets follow a step-by-step learning process that helps students better understand concepts, recognize mistakes, and possibly develop a strategy to tackle future problems. Download Multiplying and Dividing Rational Expressions Worksheet PDFsThese math worksheets should be practiced regularly and are free to download in PDF formats.
Learning ObjectivesBy the end of this section, you will be able to:
Be Prepared 7.1Before you get started, take this readiness quiz. Simplify: 90y15y2.90y15y2
.
Be Prepared 7.2Multiply: 1415·635.1415·635.
Be Prepared 7.3Divide: 1210÷825.1210÷825.
We previously reviewed the properties of fractions and their operations. We introduced rational numbers, which are just fractions where the numerators and denominators are integers. In this chapter, we will work with fractions whose numerators and denominators are polynomials. We call this kind of expression a rational expression.
Rational ExpressionA rational expression is an expression of the form pq,pq, where p and q are polynomials and q≠0.q≠0. Here are some examples of rational expressions: −24565x12y4x+1x2−94x2+ 3x−12x−8−24565x12y 4x+1x2−94x2+3x−12x−8 Notice that the first rational expression listed above, −2456−2456, is just a fraction. Since a constant is a polynomial with degree zero, the ratio of two constants is a rational expression, provided the denominator is not zero. We will do the same operations with rational expressions that we did with fractions. We will simplify, add, subtract, multiply, divide and use them in applications. Determine the Values for Which a Rational Expression is UndefinedIf the denominator is zero, the rational expression is undefined. The numerator of a rational expression may be 0—but not the denominator. When we work with a numerical fraction, it is easy to avoid dividing by zero because we can see the number in the denominator. In order to avoid dividing by zero in a rational expression, we must not allow values of the variable that will make the denominator be zero. So before we begin any operation with a rational expression, we examine it first to find the values that would make the denominator zero. That way, when we solve a rational equation for example, we will know whether the algebraic solutions we find are allowed or not.
How ToDetermine the values for which a rational expression is undefined.
Example 7.1Determine the value for which each rational expression is undefined: ⓐ 8a2b3c8 a2b3c ⓑ 4b−32b+54b−32b+5 ⓒ x+4x2+5x+6.x+4x2+5x+6 .
Try It 7.1Determine the value for which each rational expression is undefined. ⓐ 3y28x3y28 x ⓑ 8n−53n+18n−53n+1 ⓒ a+10a2+4a+3a+10a2+4a+3
Try It 7.2Determine the value for which each rational expression is undefined. ⓐ 4p5q4p5q ⓑ y−13y+2y−13y+2 ⓒ m−5m2+m−6m−5m2+m−6 Simplify Rational ExpressionsA fraction is considered simplified if there are no common factors, other than 1, in its numerator and denominator. Similarly, a simplified rational expression has no common factors, other than 1, in its numerator and denominator.
Simplified Rational ExpressionA rational expression is considered simplified if there are no common factors in its numerator and denominator. For example, x+2x+3is simplified because there are no common factors of x+2andx+3.2x3xis not simplified becausexis a common factor of2xand3x.x+2x+3is simplified because there are no common factors of x+2andx+3.2x3xis not simplified becausexis a common factor of2 xand3x. We use the Equivalent Fractions Property to simplify numerical fractions. We restate it here as we will also use it to simplify rational expressions.
Equivalent Fractions PropertyIf a, b, and c are numbers where b≠0,c≠0,b≠0,c≠0, thenab=a·cb·canda·cb·c=ab. thenab=a·cb·canda·cb·c=ab. Notice that in the Equivalent Fractions Property, the values that would make the denominators zero are specifically disallowed. We see b≠0,c≠0b≠0,c≠0 clearly stated. To simplify rational expressions, we first write the numerator and denominator in factored form. Then we remove the common factors using the Equivalent Fractions Property. Be very careful as you remove common factors. Factors are multiplied to make a product. You can remove a factor from a product. You cannot remove a term from a sum. Removing the x’s from x+5xx+5x would be like cancelling the 2’s in the fraction 2+52!2+52!
Example 7.2How to Simplify a Rational ExpressionSimplify: x2+5x+6x2+8x+12 x2+5x+6x2+8x+12.
Try It 7.3Simplify: x2−x−2x2−3x+2.x2−x−2 x2−3x+2.
Try It 7.4Simplify: x2−3x−10x2+x−2.x2−3x− 10x2+x−2. We now summarize the steps you should follow to simplify rational expressions.
How ToSimplify a rational expression.
Usually, we leave the simplified rational expression in factored form. This way, it is easy to check that we have removed all the common factors. We’ll use the methods we have learned to factor the polynomials in the numerators and denominators in the following examples. Every time we write a rational expression, we should make a statement disallowing values that would make a denominator zero. However, to let us focus on the work at hand, we will omit writing it in the examples.
Example 7.3Simplify: 3a2−12ab+12b26a2−24b2 3a2−12ab+12b26a2−24b2.
Try It 7.5Simplify: 2x2−12xy+18y23x2−27y2 2x2−12xy+18y23x2−27y2.
Try It 7.6Simplify: 5x2−30xy+25y22x2−50y2 5x2−30xy+25y22x2−50y2. Now we will see how to simplify a rational expression whose numerator and denominator have opposite factors. We previously introduced opposite notation: the opposite of a is −a−a and −a=−1·a.−a =−1·a. The numerical fraction, say 7−77−7 simplifies to −1−1. We also recognize that the numerator and denominator are opposites. The fraction a−aa −a, whose numerator and denominator are opposites also simplifies to −1−1. Let’s look at the expression b−a.b−aRewrite.−a+bFactor out–1. −1(a−b)Let’s look at the expressionb−a.b−a Rewrite.−a+bFactor out–1.−1(a−b) This tells us that b−ab−a is the opposite of a−b. a−b. In general, we could write the opposite of a−ba−b as b−a. b−a. So the rational expression a−bb−aa−bb−a simplifies to −1.−1.
Opposites in a Rational ExpressionThe opposite of a−ba−b is b−a.b−a. a−bb−a=−1a≠ba−bb−a=−1a≠b An expression and its opposite divide to −1.−1. We will use this property to simplify rational expressions that contain opposites in their numerators and denominators. Be careful not to treat a+ba+b and b+ab+a as opposites. Recall that in addition, order doesn’t matter so a+b=b+aa+b=b+a. So if a≠−b a≠−b, then a+bb+a=1.a+bb+a=1.
Example 7.4Simplify: x2−4x−3264−x2.x2 −4x−3264−x2.
Try It 7.7Simplify: x2−4x−525−x2.x2−4x−5 25−x2.
Try It 7.8Simplify: x2+x−21−x2.x2+x−21− x2. Multiply Rational ExpressionsTo multiply rational expressions, we do just what we did with numerical fractions. We multiply the numerators and multiply the denominators. Then, if there are any common factors, we remove them to simplify the result.
Multiplication of Rational ExpressionsIf p, q, r, and s are polynomials where q≠0,s≠0,q≠0,s≠0, then pq·rs=prqspq·rs=prqs To multiply rational expressions, multiply the numerators and multiply the denominators. Remember, throughout this chapter, we will assume that all numerical values that would make the denominator be zero are excluded. We will not write the restrictions for each rational expression, but keep in mind that the denominator can never be zero. So in this next example, x≠0,x≠0,x≠3,x≠3, and x≠4.x≠4.
Example 7.5How to Multiply Rational ExpressionsSimplify: 2xx2−7x+12·x2−9 6x2.2xx2−7x+12·x2−96x2.
Try It 7.9Simplify: 5xx2+5x+6·x2−410x. 5xx2+5x+6·x2−410x.
Try It 7.10Simplify: 9x2x2+11x+30·x2−363x2. 9x2x2+11x+30·x2−363x2.
How ToMultiply rational expressions.
Example 7.6Multiply: 3a2−8a−3a2−25·a2+10a+25 3a2−14a−5.3a2−8a−3a2−25·a2+10a+25 3a2−14a−5.
Try It 7.11Simplify: 2x2+5x−12x2−16·x2−8x+162x2 −13x+15.2x2+5x−12x2−16·x2−8x+162x2− 13x+15.
Try It 7.12Simplify: 4b2+7b−21−b2·b2−2b+14b2 +15b−4.4b2+7b−21−b2·b2−2b+14b2+ 15b−4. Divide Rational ExpressionsJust like we did for numerical fractions, to divide rational expressions, we multiply the first fraction by the reciprocal of the second.
Division of Rational ExpressionsIf p, q, r, and s are polynomials where q≠0,r≠0,s≠0,q≠0,r≠ 0,s≠0, then pq÷rs=pq·srpq÷rs=p q·sr To divide rational expressions, multiply the first fraction by the reciprocal of the second. Once we rewrite the division as multiplication of the first expression by the reciprocal of the second, we then factor everything and look for common factors.
Example 7.7How to Divide Rational ExpressionsDivide: p3+q32p2+2pq+2q2 ÷p2−q26.p3+q32p2+2pq+2q2÷ p2−q26.
Try It 7.13Simplify: x3−83x2−6x+12÷x2−46.x3−83x2−6x+12÷x2−46.
Try It 7.14Simplify: 2z2z2−1÷z3−z2+zz3+1. 2z2z2−1÷z3−z2+zz3+1.
How ToDivide rational expressions.
Recall from Use the Language of Algebra that a complex fraction is a fraction that contains a fraction in the numerator, the denominator or both. Also, remember a fraction bar means division. A complex fraction is another way of writing division of two fractions.
Example 7.8Divide: 6x2−7x+24x−82x2−7x+3 x2−5x+6.6x2−7x+24x−82x2−7x+ 3x2−5x+6.
Try It 7.15Simplify: 3x2+7x+24x+243x2−14x−5x2 +x−30.3x2+7x+24x+243x2−14x−5x2 +x−30.
Try It 7.16Simplify: y2−362y2+11y−62y2−2y−608y −4.y2−362y2+11y−62y2−2y−608y−4 . If we have more than two rational expressions to work with, we still follow the same procedure. The first step will be to rewrite any division as multiplication by the reciprocal. Then, we factor and multiply.
Example 7.9Perform the indicated operations: 3x−64x−4·x2+2x−3x2 −3x−10÷2x+128x+16.3x−64x−4·x2+2x−3 x2−3x−10÷2x+128x+16.
Try It 7.17Perform the indicated operations: 4m+43m−15·m2−3m−10m2−4m−32 ÷12m−366m−48.4m+43m−15·m2−3m−10m2− 4m−32÷12m−366m−48.
Try It 7.18Perform the indicated operations: 2n2+10nn−1÷n2+10n+24n2+8n −9·n+48n2+12n.2n2+10nn−1÷n2+10n+ 24n2+8n−9·n+48n2+12n. Multiply and Divide Rational FunctionsWe started this section stating that a rational expression is an expression of the form pq,pq, where p and q are polynomials and q≠0.q≠0. Similarly, we define a rational function as a function of the form R(x)= p(x)q(x)R(x)=p(x)q(x) where p(x)p(x) and q(x)q(x) are polynomial functions and q(x)q(x) is not zero.
Rational FunctionA rational function is a function of the form R(x)=p(x)q(x)R(x)=p (x)q(x) where p(x)p(x) and q(x)q(x) are polynomial functions and q(x)q(x) is not zero. The domain of a rational function is all real numbers except for those values that would cause division by zero. We must eliminate any values that make q(x)=0.q(x )=0.
How ToDetermine the domain of a rational function.
Example 7.10Find the domain of R(x)=2x2−14x4x2−16x−48. R(x)=2x2−14x4x2−16x−48.
Try It 7.19Find the domain of R(x)=2x2−10x4x2−16x−20. R(x)=2x2−10x4x2−16x−20.
Try It 7.20Find the domain of R(x)=4x2−16x8x2−16x−64. R(x)=4x2−16x8x2−16x−64. To multiply rational functions, we multiply the resulting rational expressions on the right side of the equation using the same techniques we used to multiply rational expressions.
Example 7.11Find R(x)=f(x)·g(x)R(x)= f(x)·g(x) where f(x)=2x−6x2−8x+15f(x)=2x−6x2−8x+15 and g(x)=x2−25 2x+10.g(x)=x2−252x+10.
Try It 7.21Find R(x)=f(x)·g(x)R(x)=f(x) ·g(x) where f(x)=3x−21x2−9x+14f(x )=3x−21x2−9x+14 and g(x)=2x2−83x+6 .g(x)=2x2−83x+6.
Try It 7.22Find R(x)=f(x)·g(x)R(x)=f(x) ·g(x) where f(x)=x2−x3x2+27x−30f (x)=x2−x3x2+27x−30 and g(x)=x2−100 x2−10x.g(x)=x2−100x2−10x. To divide rational functions, we divide the resulting rational expressions on the right side of the equation using the same techniques we used to divide rational expressions.
Example 7.12Find R(x)=f(x)g(x)R(x )=f(x)g(x) where f(x)=3x2x2−4x f(x)=3x2x2−4x and g(x)=9x2− 45xx2−7x+10.g(x)=9x2−45xx2−7x+10.
Try It 7.23Find R(x)=f(x)g(x)R(x)=f (x)g(x) where f(x)=2x2x2−8x f(x)=2x2x2−8x and g(x)=8x2+24xx 2+x−6.g(x)=8x2+24xx2+x−6.
Try It 7.24Find R(x)=f(x)g(x)R(x)=f (x)g(x) where f(x)=15x23x2+33xf(x)=15x23x2+33x and g(x)=5x−5x 2+9x−22.g(x)=5x−5x2+9x−22. Section 7.1 ExercisesPractice Makes PerfectDetermine the Values for Which a Rational Expression is Undefined In the following exercises, determine the values for which the rational expression is undefined. 1. ⓐ 2x2z2x2z, ⓑ 4p−16p−54p−16p −5, ⓒ n−3n2+2n−8n−3n2+2n− 8 2. ⓐ 10m11n10m11n , ⓑ 6y+134y−96y+134y−9, ⓒ b−8b2−36b−8b2−36 3. ⓐ 4x2y3y4x2y3y, ⓑ 3x−22x+13x−2 2x+1, ⓒ u−1u2−3u−28u−1u2− 3u−28 4. ⓐ 5pq29q5p q29q, ⓑ 7a−43a+57a−43a+5 , ⓒ 1x2−41x2−4 Simplify Rational Expressions In the following exercises, simplify each rational expression. 7. 8m3n12mn28m3n12mn2 8. 36v3w227vw336v3w227vw3 9. 8n−96 3n−368n−963n−36 10. 12p −2405p−10012p−2405p−100 11. x2+4x−5x2−2x +1x2+4x−5x2−2x+1 12. y2+3y−4y2−6y+5y2+3y−4y2−6y+5 13. a2 −4a2+6a−16a2−4a2+6a−16 14. y2−2y−3y2−9y2−2y−3y2−9 15. p3+ 3p2+4p+12p2+p−6p3+3p2+4p+12p2+p−6 16. x3−2x2−25x+50x2−25 x3−2x2−25x+50x2−25 17. 8b2−32b2b2−6b −808b2−32b2b2−6b−80 18. −5c2−10c−10c2+30c+100−5c2−10c−10c2+30c+ 100 19. 3 m2+30mn+75n24m2−100n23m2+30mn+75n24m 2−100n2 20. 5r2+30rs−35s2r2 −49s25r2+30rs−35s2r2−49s2 21. a−55−aa−55−a 22. 5−dd−55−dd−5 23. 20−5yy2−1620−5yy2−16 24. 4v−3264−v24v−3264−v2 25. w3+216 w2−36w3+216w2−36 26. v3+125v2−25v3+125v2−25 27. z2−9z+2016−z2 z2−9z+2016−z2 28. a 2−5a−3681−a2a2−5a−3681−a2 Multiply Rational Expressions In the following exercises, multiply the rational expressions. 29. 1216·4101216·410 30. 325·1624325·1624 31. 5x2y412xy3· 6x220y25x2y412xy3·6x220y2 32. 12a3bb2·2ab29b3 12a3bb2·2ab29b3 33. 5p2p2−5p−36· p2−1610p5p2p2−5p−36·p2−1610p 34. 3q2q2+q−6·q2−99q 3q2q2+q−6·q2−99q 35. 2y2−10yy2+10y+ 25·y+56y2y2−10yy2+10y+25·y+56y 36. z2+3zz2−3z−4·z−4z2 z2+3zz2−3z−4·z−4z2 37. 28−4b3b−3·b2 +8b−9b2−4928−4b3b−3·b2+8b−9b2−49 38. 72m−12m28m+32·m2+10m+24 m2−3672m−12m28m+32·m2+10m+24m2−36 39. 3c2 −16c+5c2−25·c2+10c+253c2−14c−53c2−16 c+5c2−25·c2+10c+253c2−14c−5 40. 2d2+d−3d2−16·d2−8d+162d2−9d−18 2d2+d−3d2−16·d2−8d+162d2−9d−18 41. 6m2−13m+2 9−m2·m2−6m+96m2+23m−46m2−13m+29 −m2·m2−6m+96m2+23m−4 42. 2n2−3n−1425−n2·n2−10n+252n2−13n+21 2n2−3n−1425−n2·n2−10n+252n2−13n+21 Divide Rational Expressions In the following exercises, divide the rational expressions. 43. v−511−v÷v2−25v−11v−511−v÷v2 −25v−11 44. 10+ww−8÷100−w2 8−w10+ww−8÷100−w28−w 45. 3s2s2−16÷s 3+4s2+16ss3−643s2s2−16÷s3+4s2+ 16ss3−64 46. r2−915÷r3− 275r2+15r+45r2−915÷r3−275r2+15r+45 47. p3+ q33p2+3pq+3q2÷p2−q212p3+q3 3p2+3pq+3q2÷p2−q212 48. v3−8w32v2+4vw+8w2÷v2−4w24 v3−8w32v2+4vw+8w2÷v2−4w24 49. x2+3x−104x÷(2x2+20x+50)x2+3x−104x÷(2x2+20x+50) 50. 2y2−10yz−48z22y−1÷( 4y2−32yz)2y2−10yz−48z22y−1÷(4y2−32 yz) 51. 2a2−a−215a+20a2+7a+12a2+8a+16 2a2−a−215a+20a2+7a+12a2+8a+16
52. 3b2+2b−812b+183b2+2b−82b2−7 b−153b2+2b−812b+183b2+2b−82b2− 7b−15 53. 12c2−122c2−3c+14c+46c2−13c+5 12c2−122c2−3c+14c+46c2−13c+5 54. 4d2+7d−235d+10d2−47d2−12d− 44d2+7d−235d+10d2−47d2−12d−4 For the following exercises, perform the indicated operations. 55. 10m2+80m3m−9·m2+4m−21m2−9m+20÷5m2+ 10m2m−1010m2+80m3m−9·m2+4m−21m2−9m+ 20÷5m2+10m2m−10 56. 4n2+32n 3n+2·3n2−n−2n2+n−30÷108n2−24nn+6 4n2+32n3n+2·3n2−n−2n2+n−30÷108n2−24nn+6 57. 12p 2+3pp+3÷p2+2p−63p2−p−12·p−79p3−9p2 12p2+3pp+3÷p2+2p−63p2−p−12·p−79p 3−9p2 58. 6q+39q2−9q÷q 2+14q+33q2+4q−5·4q2+12q12q+66q+39q 2−9q÷q2+14q+33q2+4q−5·4q2+12q12q+6 Multiply and Divide Rational Functions In the following exercises, find the domain of each function. 59. R(x)=x3−2x2−25x+50x2−25R(x)= x3−2x2−25x+50x2−25 60. R(x) =x3+3x2−4x−12x2−4R(x)=x3+3x2−4x −12x2−4 61. R(x)=3x2+15x6x2+6x−36R(x)=3x2 +15x6x2+6x−36 62. R(x)=8x2 −32x2x2−6x−80R(x)=8x2−32x2x2−6x−80 For the following exercises, find R(x)=f(x)·g(x)R(x)=f( x)·g(x) where f(x)f(x) and g( x)g(x) are given. 63. f(x)=6x2−12xx2+7x−18f(x)=6
x2−12xx2+7x−18 64. f(x)=x2−2xx2+6x−16f(x)
=x2−2xx2+6x−16 65. f(x)=4xx
2−3x−10f(x)=4xx2−3x−10 66. f(x)=2x2+8xx2−9x+20
f(x)=2x2+8xx2−9x+20 For the following exercises, find R(x)=f(x)g(x)R(x)=f( x)g(x) where f(x)f(x) and g (x)g(x) are given. 67. f(x)=27x23x
−21f(x)=27x23x−21 68. f(x)=24x22
x−8f(x)=24x22x−8 69. f(x)=16x24x
+36f(x)=16x24x+36 70. f(x)=24x22x
−4f(x)=24x22x−4 Writing Exercises71. Explain how you find the values of x for which the rational expression x2−x− 20x2−4x2−x−20x2−4 is undefined. 72. Explain all the steps you take to simplify the rational expression p2+4p−219−p2.p2+4p−21 9−p2. 73. ⓐ Multiply 74·91074·910 and explain all your steps. ⓑ Multiply nn− 3·9n+3nn−3·9n+3 and explain all your steps. ⓒ Evaluate your answer to part ⓑ when n=7n=7. Did you get the same answer you got in part ⓐ? Why or why not? 74. ⓐ Divide 245 ÷6245÷6 and explain all your steps. ⓑ Divide x2−1x÷(x+1) x2−1x÷(x+1) and explain all your steps. ⓒ Evaluate your answer to part ⓑ when x=5. x=5. Did you get the same answer you got in part ⓐ? Why or why not? Self Checkⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. ⓑ If most of your checks were: …confidently. Congratulations! You have achieved your goals in this section! Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific! …with some help. This must be addressed quickly as topics you do not master become potholes in your road to success. Math is sequential - every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help?Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved? …no - I don’t get it! This is critical and you must not ignore it. You need to get help immediately or you will quickly be overwhelmed. See your instructor as soon as possible to discuss your situation. Together you can come up with a plan to get you the help you need. How do you divide rational expressions step by step?How to divide rational expressions.. Rewrite the division as the product of the first rational expression and the reciprocal of the second.. Factor the numerators and denominators completely.. Multiply the numerators and denominators together.. Simplify by dividing out common factors.. How do you multiply rational expressions?To multiply rational expressions: Completely factor all numerators and denominators. Reduce all common factors. Either multiply the denominators and numerators or leave the answer in factored form.
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Multiplying and dividing rational expressions answer key
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