Chapter 1D - Rational Expressions
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1 - Capter 1D Capter 1D - Rational Expressions Definition of a Rational Expression A rational expression is te quotient of two polynomials. (Recall: A function px is a polynomial in x of degree n, if tere are constants a 0, a 1,, a n,wita n 0 suc tat px a 0 a 1 x a n x n.) More formally, a rational expression is an expression of te form p q, were p and q are polynomials, and qx cannot be te zero polynomial. Te denominator of a rational expression can be a constant polynomial toug. For example, rx x is a polynomial of degree 1, and it is also a rational expression, for rx x 1 x. Example 1: 4x 7 6x 16, x 9, x 3 8, and x 5 x 5 are examples of rational expressions. Example : x 1 x 7 is not a rational expression. A rational expression involves a quotient, and since division by 0 is not defined for real numbers, tere are sometimes restrictions on te variable x. In particular, te restrictions occur werever te denominator is zero. Example 3: Forte rational expression 6x 4x 16 7,tevaluesofxwic make te denominator zero can be found by solving te equation 6x x 16 x Terefore, te variable x cannot equal 8 in te rational expression 4x 7 3 6x 16 domain of tis expression is all x not equal to We also say tat te Example 4: For te rational expression,tevaluesofxwic make te denominator zero x 9 can be found by solving te equation x 9 0 x 9 Terefore, te variable x cannot equal 9 in te rational expression x 9.Tedomainoftis expression is all x not equal to 9. Example 5: For te rational expression x 5,terearenovaluesofxwic make te x 5 denominator zero since x 5 is always positive. Terefore, tere are no restrictions on te variable x. Te domain of tis expression is all x.
2 - Capter 1D Example 6: For te rational expression x 9,tevaluesofxwic make te denominator zero x 1 can be found by solving te equation x 1 0 x 1 x 1 Terefore, te variable x cannot equal 1 in te rational expression x 9 x 1.Tedomainoftis expression is all x not equal to 1.
3 - Capter 1D Simplifying Rational Expressions To simplify a rational expression means to reduce it to lowest terms. From working wit fractions, you may recall tat simplifying is done by cancelling common factors. Terefore, te key to simplifying rational expressions (and to most problems involving rational expressions) is to factor te polynomials wenever possible. Example 1: Write 8x4 in reduced form. 1x 6 Solution: Factor te numbers and cancel common factors. Use properties of exponents to elp cancel te x s: 8x 4 4 x 4 1x x 4 x 3x Example : Solution: Write y3 5y 6y y 4 y 3 5y 6y y 4 in reduced form. yy 5y 6 y 4 yy y 3 y y yy 3 y y 3y y ; y Question: Wy is y? Answer: In order for our answer to be equivalent to te original fraction, te variable must ave te same restrictions. Since y cannot equal inteoriginalexpression(tedenominatorwouldten be zero), we must restrict te domain of our answer in order for tese fractions to be equivalent. Example 3: Write 4x x 3 x x in reduced form. Solution: 4x x 3 x x x4 x x x x x x x x 1 At tis point, it doesn t look like anyting will cancel. However, if we factor 1 fromtelasttermin te numerator, we obtain te following: x xx x x 1 xx x 1 x x x 1 ; x or 1 Question: Answer: Wat property ofreal numbers tells us tat x x xx? Te commutative property of addition. x x.
4 - Capter 1D Operations wit Rational Expressions Te operations of addition, subtraction, multiplication, and division wit rational expressions follow te same rules tat are used wit common fractions. Te key, as wit simplifying rational expressions, is to factor te polynomials. I. Multiplication of Rational Expressions Recall wit fractions a b and d c tat a b d c ac. Also recall wen mulitplying fractions tat you bd may cross-cancel; tat is, reduce common factors of any numerator wit any denominator. Example 1: Te same is true of multiplying rational expressions. As before, te key is to factor te polynomials first. Example : Simplify 5 x 1 x 1 5x 50 Solution: First factor te polynomials. Don t forget to cancel common factors! 5 x 1 x 1 5x 50 5 x 1 x 1x 1 5x 1 1 x 1 5x x 1 5x 10 ; x 1or Before looking at te next example, recall our strategies for factoring: 1. Factor out any common factors. If more tan 3 terms, try factoring by grouping 3. Recognize special products 4. Factor trinomials using product/sum strategies: a) x bx c: factors into x mx n were mn c and m n b b) ax bx c: split bx into mx nx, were mn ac and m n b, tenfactorbygrouping. Example 3: Simplify x x 6 x 4x 5 x3 3x x x 5 4x 6x x 3 8 Solution: x x 6 x 4x 5 x3 3x x x 5 x 3x x 5x 1 Example 4: Simplify x 4 x 4 x 3 8 4x 6x xx x 1 xx 3 x 5 x x x 4 x x x 4 x x 4x 8 ; x 5,1,0, 3,
5 - Capter 1D Solution: x 4 x 4 x x x x, x. II. Division of Rational Expressions Recall wit fractions a b and d c tat a b d c a/b c/d a b d c. In oter words, dividing by a fraction is te same as multiplying by its reciprocal. Te same is true for rational expressions. As before, te key to making te work easier is to factor te polynomials first. Example 5: x x 15 x x 7x 10 Solution: Begin by writing te second term as x 7x 10 : x x 15 x x 7x x x 15 1 x x 7x 10 x 5x 3 1 x x 5x x 3 x x 3 x 4x 4 ; x 5, Notice tat te x terms cannot be cancelled since bot are in te denominator. Multiplication and Division can also appear togeter. Only take te reciprocal ofte fractions you are dividing. y Example 6: 7y 1 y 3y 18 y 3y 8 3y 1 y 1y 36 4y 4 Solution: We only take te reciprocal of te second fraction: y 7y 1 y 3y 18 y 3y 8 3y 1 y 1y 36 4y 4 y 7y 1 y 3y 18 y 1y 36 y 3y 8 y 3y 4 y 6y 3 y 6 y 7y 4 3; y 7, 6, 3, 4 3y 1 4y 4 3y 7 4y 6 Question: Explain wy te values 7, 6, 3, and 4 are excluded in te previous example. Answer: Te easiest way to understand wy tese values ave been excluded is to write y 7y 1 y 3y 18 y 3y 8 3y 1 as a fraction. y 1y 36 4y 4 y 7y 1 y 3y 18 3y 1 y 3y 8 4y 4 y 1y 36 y 6followssince4y 4 cannot equal 0. We also get y 3becauseteterm y 7y 1 y 3y 18 y 7y 1 y 6y 3
6 - Capter 1D Te next observation is tat te denominator of y 7y 1 y 3y 18 y 3y 8 y 1y 36 y tat is, 3y 8 cannot equal zero. Factoring te numerator and denominator of tis rational y 1y 36 expression we ave y 3y 8 y 7y 4 Te numerator is zero if y 7 orify 4. Tis accounts for excluding te numbers 7 and 4. Te denominator of y 3y 8 y 1y 36 factors into y 6 wic means we ave to exclude 6, but we ave already done tat. Question: Wat values ofx are not allowed in te rational expression x x 1 6 x x 3 4? Answer: 3, 4, 6 (Te value 3 is not allowed because x x 3 is zero at x 3 and we cannot 4 divide by 0. x 4isnotallowedbecauseweavetetermx 4intedenominator. Similarly x 6is not allowed because te term x 6istedenominatorof x x 1 6.) III. Addition and Subtraction of Rational Expressions Recall tat addition and subtraction is done by first finding a common denominator. Te least common denominator (LCD) of several fractions is te product of all prime factors of te denominators. A factor only occurs more tan once in te LCD if it occurs more tan once in any one fraction. Once you ave te LCD, convert eac fraction to an equivalent fraction wit te LCD, ten add or subtract te numerators. Example 7: 1 x 1 x 1 Te LCD ere is xx 1. We convert eac fraction to an equivalent one wit tis x xx 1 Solution: denominator: Tat is, 1 x x 1 xx 1 and 1 x 1 1 x 1 x 1 1x 1 xx 1 x 1 xx 1 x 1 x xx 1 1x x 1x x xx 1 1 xx 1 Example 8: Solution: y 4 y 6 y 3y 8 y 1y 36 We must factor te denominators first to find te LCD:
7 - Capter 1D y 4 y 6 y 3y 8 y 6 Te LCD is y 6. We now convert eac fraction to an equivalent fraction using tis denominator: y 4 y 4y 6 y 6 y 6 y 3y 8 y 3y 8 y 6 y 6 Next subtract te two expressions y 4 y 6 y 3y 8 y 1y 36 y 4y 6 y 3y 8 y 6 y 6 y y 4 y 3y 8 y 6 y 6 y y 4 y 3y 8 y 6 y y 4 y 3y 8 y 6 y 4 y 6 Note tat te minus sign in te numerator was distributed across te expression y 3y 8 y 3y 8. Question: Wat is te common denominatorin x 5 x 7x 1 1 x 4? Answer: x 7x 1x 4
8 - Capter 1D Compound Fractions A compound fraction (also called a complex fraction) is an expression wic contains nested fractions. In general, tere is one main fraction wic will ave one or more fractions in te numerator and/or denominator. Tere are two strategies wic may be used to simplify compound fractions. First, simplify bot te numerator and te denominator individually, ten divide te numerator by te denominator by multiplying wit te reciprocal of te denominator. 6 y 5 y 1 Example 1: 6 y 4 Solution: Example First simplify te numerator using te common denominator yy 1: 6 y 5 y 1 6 y 4 6y 1 yy 1 6 y 4 5y y 1y 1y 6 5y yy 1 6 y 4 7y 6 yy 1 6 y 4 Now simplify te denominator using te common denominator y (remember tat : 6 y 5 y 1 6 y 4 7y 6 yy 1 6 y 4y 1y We now divide te two fractions 6 y 5 y 1 6 y 4 7y 6 yy 1 7y 6 yy 1 6 4y y 6 4y and y : 7y 6 yy 1 7y 6 yy 1 6 4y y y 6 4y 7y 6 y 16 4y ; y 3, 1,0. As you can see, tis is often a long, tedious process. A second tecnique is to multiply te numerator and te denominator of te main fraction by te LCD of all te smaller fractions and simplify te result.
9 - Capter 1D Example : 6 y 5 y 1 6 y 4 Solution: Te LCD of all te smaller fractions is yy 1. We multiply te numerator and denominator by tis LCD: 6 y 5 y 1 6 y 4 6 y 5 yy 1 y 1 6 y 4 yy 1 6 y yy 1 5 yy 1 y 1 6 y yy 1 4yy 1 6y 1 5y 6y 1 4yy 1 1y 6 5y y 16 4y 7y 6 y 16 4y Te following example is a common one in many calculus classes: Example 3: Simplify te expression x 1 1 x 1 1. Solution: First rewrite te expression witout negative exponents. (Recall x 1 1 x ) x 1 1 x 1 1 x 1 x 1 Metod I Solution: Now simplify te numerator using te common denominator x 1x 1: x 1 x 1 1 x 1 1 x 1x 1 x 1 x 1x 1 x x x 1x 1 x x x 1x 1 x 1x 1 Te denominator is wic is already simplified. We now divide te two fractions 1 x 1x 1 and 1 : x 1 1 x 1 1 x 1x 1 1 x 1x 1 ; 0
10 - Capter 1D Metod II Solution: Te LCD of all te smaller fractions is x 1x 1. We mulitply te numerator and denominator by tis LCD: x 1 x 1 x 1x 1 x 1x 1 x 1 x 1x 1 x 1 x 1x 1 x 1 x 1 x 1x 1 x x x 1x 1 x 1x 1 x 1x 1 ; 0 x 1x 1
11 - Capter 1D Exercises for Capter 1D - Rational Expressions For problems 1-5, determine if te given expression is a rational expression. If it is a rational expression, state any restrictions on te variable. 1. 3x 6x 7. 5x 10 5x x 8 x 4x x x 4 For problems 6-8, reduce te rational expression if possible. x 6. 3 y x y 7. x 4x 1 5x x 3 4x x x For problems 9-0, perform te indicated operations and simplify. 9. x 3 8 x 4 x 3 8 x x z 5z 4 9z 1z 4 z 9 z 10z 16 z z 6 9z x 1 6x 9 x 9x 8 1. y 8y 0 y 11y 10 y3 1 y 3 8 y y 1 y y y 3 7 4y y 6 y 3 y 5y 6 y y 14. x x 3 3x x 3 x x 3 1 x 3 10 x x 1 8x x x 1 x x 3 x y 7y 1 y 3y 18 y 5 (HINT: Simplify first) y 8y y 7 y y 1 5 y 1 1. For problems 1-5, simplify te expression completely. 3x y 4 x y
12 - Capter 1D 5. x 1 x x 1 x a 1 b x 1 x x x
13 - Capter 1D Answers to Exercises for Capter 1D - Rational Expressions 1. Tis is a rational function and tere are no restrictions on te variable x.. Is a rational function Restriction: 5x x 10 x 3. Not a rational function because ofte radical in te numerator. 4. Tis is a rational function and tere are no restrictions on te variable x. 5. Is a rational function. Restriction: x x 3 4x x x x 3 y x y x 4x 1 5x 10 x 4 x x y x y y x y ; x 0 and y 0 xx x x x 1 x 6x 5x x 3 8 x 4 x 3 8 x x 4 x x x 4 x x xx x 1 x 6 5 ; x ; x and x 1 x x x 4 x x 4 z 5z 4 9z 1z 4 z 9 z 10z 16 z z 6 9z 1 z 8z 3 z 3z 3 3z z 3z z z 8 33z 4 1 3z z 3 z 3z 4 z 3,3,8,, x 1 6x 9 x 9x 8 3 x 1 x 9x 8 6x 9 3 x 1 x 8x 1 3x 3 x x 4; x, x 8 x 3 ; x 1,8
14 - Capter 1D LCD: x 33x 4 x x 3 3x LCD: x x 3 1 x 3 x LCD: x 3x y 8y 0 y 11y 10 y3 1 y 3 8 y y 1 y y 4 y 10y y 10y 1 1; y 10, 1, y 3 7 y 3 y 1y y 1 y y y 4 y y 4 y y 1 4y y 5y 6 y 3y 3y 9 y 3 y 3y 9 yy 3 ; y x3x 4 x 33x 4 1x 3 x x 3 LCD: x 1x 1 LCD: xx 1x 1 1x 3 x 3x 3 x 3 x 3 10 x 3x 3 x 10 x 3x 3 7x 1 x 1x 1 y 6 y y 4y y y 3 yy y 3 x 3 3x 4x 3 3x 4x x 6 x 33x 4 3x x 3x 7 x 1 x 3 3x 6 x x 3 1x 3 x 3x 3 10 x 3x 3 8x x xx 1 x 1x 1 14x 7 16x 8x 16x 4 x 1x 1 6x 11 x 1x 1 3 x 1 x x 3 x 1 3 x 1 x x 3 x 1x 1 4x 1x 1 x 1x 1 3x 6x 6 x 33x 4 x 9 x x 3
15 - Capter 1D So LCD is y 6y 3 So LCD y 1 3xx 1 x 1xx 1 x 1x 1 xx 1x 1 3x 3x x x 3x xx 1x 1 x 6x xx 1x 1 y 7y 1 y 3y 18 y 5 y 8y 15 y 4y 3 y 5y 5 y 6y 3 y 5y 3 y 4 y 6 y 5 y 3 ; y 5 y 4y 3 y 6y 3 y 5y 6 y 3y 6 y 7y 1 y 11y 30 y 6y 3 y 4y 4 y 6y 3 ; y 5 y 7 y y 1 5 y 1 y 7 y 1 5 y 1 y 7 y 1 5y 1 y 1 y 7 5y 5 y 1 4y 1 y 1 3x y 4 4 x y 4 3x y x y 3 x 3x x 1x 1x
16 - Capter 1D x 1 x x 1 x x 1 x x 1 x 1x 1 5x 1 xx 1 x 1x 1 5x 5 x x x 1 x 6x 5 x 1 ; x 1, 1 x 1x 1 1 x 1x 1 1ab 1 a 1 ab b ab b a ; a 0;b 0 1 x 1 5. x 1 x 1 x x x 1 x xx x xx 1 x x x 1 x 1 x x x x x x x x x x x x x x x x x x x x x x x x x x x x ; 0
1 1. Rationalize the denominator and fully simplify the radical expression 3 3. Solution: = 1 = 3 3 = 2
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