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College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson P Prerequisites P.7 Rational Expressions Fractional Expression A quotient of two algebraic expressions is called a fractional expression. • Here are some examples: 3 x x x 2x y 2 x 1 y 2 4 x 2 5x 6 x 2 1 Rational Expression A rational expression is a fractional expression in which both the numerator and denominator are polynomials. • Here are some examples: 2x x 1 y 2 y2 4 x x x 2 5x 6 3 Rational Expressions In this section, we learn: • How to perform algebraic operations on rational expressions. The Domain of an Algebraic Expression The Domain of an Algebraic Expression In general, an algebraic expression may not be defined for all values of the variable. The domain of an algebraic expression is: • The set of real numbers that the variable is permitted to have. The Domain of an Algebraic Expression The table gives some basic expressions and their domains. E.g. 1—Finding the Domain of an Expression Find the domains of these expressions. 2 a 2 x 3x 1 x b 2 x 5x 6 x c x 5 E.g. 1—Finding the Domain Example (a) 2x2 + 3x – 1 This polynomial is defined for every x. • Thus, the domain is the set of real numbers. E.g. 1—Finding the Domain Example (b) We first factor the denominator: x x 2 x 5 x 6 x 2 x 3 • Since the denominator is zero when x = 2 or x = 3. • The expression is not defined for these numbers. • The domain is: {x | x ≠ 2 and x ≠ 3}. E.g. 1—Finding the Domain Example (c) x x 5 • For the numerator to be defined, we must have x ≥ 0. • Also, we cannot divide by zero, so x ≠ 5. • Thus the domain is {x | x ≥ 0 and x ≠ 5}. Simplifying Rational Expressions Simplifying Rational Expressions To simplify rational expressions, we factor both numerator and denominator and use following property of fractions: AC A BC B • This allows us to cancel common factors from the numerator and denominator. E.g. 2—Simplifying Rational Expressions by Cancellation x 1 Simplify: 2 x x 2 2 x 1 x2 x 2 x 1 x 1 x 1 x 2 2 x 1 x2 (Factor) (Cancel common factors) Caution We can’t cancel the x2’s in x 1 2 x x 2 2 because x2 is not a factor. Multiplying and Dividing Rational Expressions Multiplying Rational Expressions To multiply rational expressions, we use the following property of fractions: A C AC B D BD This says that: • To multiply two fractions, we multiply their numerators and multiply their denominators. E.g. 3—Multiplying Rational Expressions Perform the indicated multiplication, and simplify: x 2x 3 3 x 12 2 x 8 x 16 x 1 2 E.g. 3—Multiplying Rational Expressions We first factor: x 2 2 x 3 3 x 12 2 x 8 x 16 x 1 x 1 x 3 3 x 4 Factor 2 x 1 x 4 3 x 1 x 3 x 4 x 1 x 4 3 x 3 x4 2 Property of fractions Cancel common factors Dividing Rational Expressions To divide rational expressions, we use the following property of fractions: A C A D B D B C This says that: • To divide a fraction by another fraction, we invert the divisor and multiply. E.g. 4—Dividing Rational Expressions Perform the indicated division, and simplify: x 4 x 3x 4 2 2 x 4 x 5x 6 2 E.g. 4—Dividing Rational Expressions x 4 x 2 3x 4 2 2 x 4 x 5x 6 x 4 x 2 5x 6 2 2 Invert diviser and multiply x 4 x 3x 4 x 4 x 2 x 3 Factor x 2 x 2 x 4 x 1 x 3 x 2 x 1 Cancel common factors Adding and Subtracting Rational Expressions Adding and Subtracting Rational Expressions To add or subtract rational expressions, we first find a common denominator and then use the following property of fractions: A B AB C C C Adding and Subtracting Rational Expressions Although any common denominator will work, it is best to use the least common denominator (LCD) as explained in Section P.2. • The LCD is found by factoring each denominator and taking the product of the distinct factors, using the highest power that appears in any of the factors. Caution Avoid making the following error: A A A B C B C Caution For instance, if we let A = 2, B = 1, and C = 1, then we see the error: 2 ?2 2 1 1 1 1 2? 2 2 2 ? 1 4 Wrong! E.g. 5—Adding and Subtracting Rational Expressions Perform the indicated operations, and simplify: 3 x (a) x 1 x 2 1 2 (b) 2 2 x 1 x 1 E.g. 5—Adding Rational Exp. Example (a) Here LCD is simply the product (x – 1)(x + 2). 3 x x 1 x 2 3 x 2 x x 1 x 1 x 2 x 1 x 2 Fractions by LCD 3x 6 x x x 1 x 2 Add fractions x 2x 6 x 1 x 2 Combine terms in numerator 2 2 E.g. 5—Subtracting Rational Exp. Example (b) The LCD of x2 – 1 = (x – 1)(x + 1) and (x + 1)2 is (x – 1)(x + 1)2. 1 2 2 2 x 1 x 1 1 2 x 1 x 1 x 1 x 1 2 x 1 2 x 1 x 1 2 Factor Combine fractions using LCD E.g. 5—Subtracting Rational Exp. x 1 2x 2 x 1 x 1 2 3x x 1 x 1 2 Example (b) Distributive Property Combine terms in numerator Compound Fractions Compound Fraction A compound fraction is: • A fraction in which the numerator, the denominator, or both, are themselves fractional expressions. E.g. 6—Simplifying a Compound Fraction Simplify: x 1 y y 1 x E.g. 6—Simplifying Solution 1 One solution is as follows. 1. We combine the terms in the numerator into a single fraction. 2. We do the same in the denominator. 3. Then we invert and multiply. E.g. 6—Simplifying Thus, Solution 1 x xy 1 y y y xy 1 x x xy x y xy x x y y x y E.g. 6—Simplifying Solution 2 Another solution is as follows. 1. We find the LCD of all the fractions in the expression, 2. Then multiply the numerator and denominator by it. E.g. 6—Simplifying Solution 2 Here, the LCD of all the fractions is xy. x x 1 1 xy y y y y xy 1 1 x x 2 x xy 2 xy y x x y y x y Multiply num. and denom.by xy Simplify Factor Simplifying a Compound Fraction The next two examples show situations in calculus that require the ability to work with fractional expressions. E.g. 7—Simplifying a Compound Fraction 1 1 Simplify: ah a h • We begin by combining the fractions in the numerator using a common denominator: E.g. 7—Simplifying a Compound Fraction 1 1 ah a h a a h a a h h a a h 1 a a h h Combine fractions in numerator Invert divisor and multiply E.g. 7—Simplifying a Compound Fraction aah 1 a a h h DistributiveProperty h 1 a a h h Simplify 1 a a h Cancel common factors E.g. 8—Simplifying a Compound Fraction Simplify: 1 x 1 2 2 x 1 x 2 1 x 2 1 2 2 E.g. 8—Simplifying Solution 1 Factor (1 + x2)–1/2 from the numerator. 1 x 2 1/2 x 1 x 2 1 x 2 2 1/2 1 x 2 1 x 2 1/2 1/2 1 x 2 1 1 x 2 3/2 1 x 2 x 2 1 x 2 E.g. 8—Simplifying Solution 2 Since (1 + x2)–1/2 = 1/(1 + x2)1/2 is a fraction, we can clear all fractions by multiplying numerator and denominator by (1 + x2)1/2. E.g. 8—Simplifying Solution 2 Thus, 1 x 2 1/2 2 1 x 2 x 1 x 1 x 2 1/2 2 1 x 2 1 x x 1 x 1 x 2 x 2 2 2 3/2 1/2 2 1/2 1 x 1 x 2 2 1 1 x 2 3/2 1/2 1/2 Rationalizing the Denominator or the Numerator Rationalizing the Denominator If a fraction has a denominator of the form AB C we may rationalize the denominator by multiplying numerator and denominator by the conjugate radical A B C . Rationalizing the Denominator This is effective because, by Product Formula 1 in Section P.5, the product of the denominator and its conjugate radical does not contain a radical: A B C A B C A 2 B C 2 E.g. 9—Rationalizing the Denominator Rationalize the denominator: 1 1 2 • We multiply both the numerator and the denominator by the conjugate radical of 1 2 , which is 1 2 . E.g. 9—Rationalizing the Denominator Thus, 1 1 2 1 1 2 1 2 1 2 1 2 1 2 2 2 Product Formula1 1 2 1 2 2 1 1 2 1 E.g. 10—Rationalizing the Numerator Rationalize the numerator: 4h 2 h • We multiply numerator and denominator by the conjugate radical 4 h 2: E.g. 10—Rationalizing the Numerator Thus, 4h 2 h 4h 2 4h 2 h 4h 2 h 4h 2 2 2 4h 2 Product Formula1 E.g. 10—Rationalizing the Numerator h h 4h4 4h 2 h 4h 2 1 4h 2 Cancel common factors Avoiding Common Errors Avoiding Common Errors Don’t make the mistake of applying properties of multiplication to the operation of addition. • Many of the common errors in algebra involve doing just that. Avoiding Common Errors The following table states several multiplication properties and illustrates the error in applying them to addition. Avoiding Common Errors To verify that the equations in the right-hand column are wrong, simply substitute numbers for a and b and calculate each side. Avoiding Common Errors For example, if we take a = 2 and b = 2 in the fourth error, we get different values for the left- and right-hand sides: Avoiding Common Errors 1 1 1 1 The left-hand side is: 1 a b 2 2 1 1 1 The right-hand side is: ab 22 4 • Since 1 ≠ ¼, the stated equation is wrong. Avoiding Common Errors You should similarly convince yourself of the error in each of the other equations. • See Exercise 119.