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Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 8. 2 - 1

Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 8. 2 - 1

Chapter 8 Rational Expressions and Functions Copyright © 2010 Pearson Education, Inc. All rights

Chapter 8 Rational Expressions and Functions Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 8. 2 - 2

8. 2 Adding and Subtracting Rational Expressions Copyright © 2010 Pearson Education, Inc. All

8. 2 Adding and Subtracting Rational Expressions Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 8. 2 - 3

8. 2 Adding and Subtracting Rational Expressions Objectives 1. Add and subtract rational expressions

8. 2 Adding and Subtracting Rational Expressions Objectives 1. Add and subtract rational expressions with the same denominator. 2. Find a least common denominator. 3. Add and subtract rational expressions with different denominators. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8. 2 - 4

8. 2 Adding and Subtracting Rational Expressions Adding or Subtracting Rational Expressions Step 1

8. 2 Adding and Subtracting Rational Expressions Adding or Subtracting Rational Expressions Step 1 If the denominators are the same, add or subtract the numerators. Place the result over the common denominator. If the denominators are different, first find the least common denominator. Write all rational expressions with this LCD, and then add or subtract the numerators. Place the result over the common denominator. Step 2 Simplify. Write all answers in lowest terms. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8. 2 - 5

8. 2 Adding and Subtracting Rational Expressions EXAMPLE 1 Adding and Subtracting Rational Expressions

8. 2 Adding and Subtracting Rational Expressions EXAMPLE 1 Adding and Subtracting Rational Expressions with the Same Denominator Add or subtract as indicated. (a) 4 m 5 n + 7 7 = 4 m + 5 n 7 (b) 1 – 5 g 3 = – 43 g Copyright © 2010 Pearson Education, Inc. All rights reserved. Add the numerators. Keep the common denominator. Subtract the numerators; keep the common denominator. Simplify. Sec 8. 2 - 6

8. 2 Adding and Subtracting Rational Expressions EXAMPLE 1 Adding and Subtracting Rational Expressions

8. 2 Adding and Subtracting Rational Expressions EXAMPLE 1 Adding and Subtracting Rational Expressions with the Same Denominator Add or subtract as indicated. (c) a a 2 – b 2 = a–b (a – b)(a + b) Factor. = 1 a+b Lowest terms Copyright © 2010 Pearson Education, Inc. All rights reserved. Subtract the numerators; keep the common denominator. Sec 8. 2 - 7

8. 2 Adding and Subtracting Rational Expressions EXAMPLE 1 Adding and Subtracting Rational Expressions

8. 2 Adding and Subtracting Rational Expressions EXAMPLE 1 Adding and Subtracting Rational Expressions with the Same Denominator Add or subtract as indicated. (d) 5 k 2 + 2 k – 15 + k k 2 + 2 k – 15 = 5+k k 2 + 2 k – 15 Add. = 5+k (k – 3)(k + 5) Factor. = 1 k– 3 Copyright © 2010 Pearson Education, Inc. All rights reserved. Lowest terms Sec 8. 2 - 8

8. 2 Adding and Subtracting Rational Expressions The Least Common Denominator Finding the Least

8. 2 Adding and Subtracting Rational Expressions The Least Common Denominator Finding the Least Common Denominator Step 1 Factor each denominator. Step 2 Find the least common denominator. The LCD is the product of all different factors from each denominator, with each factor raised to the greatest power that occurs in the denominator. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8. 2 - 9

8. 2 Adding and Subtracting Rational Expressions EXAMPLE 2 Finding Least Common Denominators Assume

8. 2 Adding and Subtracting Rational Expressions EXAMPLE 2 Finding Least Common Denominators Assume that the given expressions are denominators of fractions. Find the LCD for each group. (a) 4 m 3 n 2, 6 m 2 n 5 Factor each denominator. 22 · m 3 · n 2 2 · 3 · m 2 · n 5 LCD = = 22 · 3 · m 3 · n 5 Choose the factors with the greatest exponents. 12 m 3 n 5 Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8. 2 - 10

8. 2 Adding and Subtracting Rational Expressions EXAMPLE 2 Finding Least Common Denominators Assume

8. 2 Adding and Subtracting Rational Expressions EXAMPLE 2 Finding Least Common Denominators Assume that the given expressions are denominators of fractions. Find the LCD for each group. (b) y – 5, y Each denominator is already factored. The LCD, an expression divisible by both y – 5 and y is y(y – 5). It is usually best to leave a least common denominator in factored form. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8. 2 - 11

8. 2 Adding and Subtracting Rational Expressions EXAMPLE 2 Finding Least Common Denominators Assume

8. 2 Adding and Subtracting Rational Expressions EXAMPLE 2 Finding Least Common Denominators Assume that the given expressions are denominators of fractions. Find the LCD for each group. (c) n 2 – 3 n – 10, n 2 – 8 n + 15 Factor the denominators. n 2 – 3 n – 10 = (n – 5)(n + 2) n 2 – 8 n + 15 = (n – 5)(n – 3) Factor. The LCD, divisible by both polynomials, is (n – 5)(n + 2)(n – 3). Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8. 2 - 12

8. 2 Adding and Subtracting Rational Expressions EXAMPLE 2 Finding Least Common Denominators Assume

8. 2 Adding and Subtracting Rational Expressions EXAMPLE 2 Finding Least Common Denominators Assume that the given expressions are denominators of fractions. Find the LCD for each group. (d) 4 h 2 – 12 h, 3 h – 9 4 h 2 – 12 h = 4 h(h – 3) 3 h – 9 = 3(h – 3) Factor. The LCD is 4 h· 3·(h – 3) = 12 h(h – 3). Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8. 2 - 13

8. 2 Adding and Subtracting Rational Expressions EXAMPLE 2 Finding Least Common Denominators Assume

8. 2 Adding and Subtracting Rational Expressions EXAMPLE 2 Finding Least Common Denominators Assume that the given expressions are denominators of fractions. Find the LCD for each group. (e) g 2 – 2 g + 1, g 2 + 3 g – 4, 5 g + 20 g 2 – 2 g + 1 = (g – 1)2 g 2 + 3 g – 4 = (g – 1)(g + 4) Factor. 5 g + 20 = 5(g + 4) The LCD is 5(g – 1)2(g + 4). Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8. 2 - 14

8. 2 Adding and Subtracting Rational Expressions EXAMPLE 3 Adding and Subtracting Rational Expressions

8. 2 Adding and Subtracting Rational Expressions EXAMPLE 3 Adding and Subtracting Rational Expressions with Different Denominators Add or subtract as indicated. The LCD of 3 z and 9 z is 9 z. (a) 7 1 + 3 z 9 z = 7· 3 3 z · 3 + 1 9 z = 21 9 z + 1 9 z = 21 + 1 9 z = 22 9 z Copyright © 2010 Pearson Education, Inc. All rights reserved. Fundamental property Add the numerators. Sec 8. 2 - 15

8. 2 Adding and Subtracting Rational Expressions EXAMPLE 3 Adding and Subtracting Rational Expressions

8. 2 Adding and Subtracting Rational Expressions EXAMPLE 3 Adding and Subtracting Rational Expressions with Different Denominators Add or subtract as indicated. The LCD is y(y – 4). (b) 2 y – 3 y– 4 = 2(y – 4) y(y – 4) – y· 3 y(y – 4) Fundamental property = 2 y – 8 y(y – 4) – 3 y y(y – 4) Distributive and commutative properties = 2 y – 8 – 3 y y(y – 4) = –y – 8 y(y – 4) Copyright © 2010 Pearson Education, Inc. All rights reserved. Subtract the numerators. Combine like terms in the numerator. Sec 8. 2 - 16

8. 2 Adding and Subtracting Rational Expressions Caution with Subtracting Rational Expressions CAUTION One

8. 2 Adding and Subtracting Rational Expressions Caution with Subtracting Rational Expressions CAUTION One of the most common sign errors in algebra occurs when a rational expression with two or more terms in the numerator is being subtracted. In this situation, the subtraction sign must be distributed to every term in the numerator of the fraction that follows it. Carefully study the example below to see how this is done. 8 d d+5 – d– 6 d+5 = 8 d – (d – 6) d+5 Combine terms in the numerator. Copyright © 2010 Pearson Education, Inc. All rights reserved. Subtract the numerators; keep the common denominator. 8 d – d + 6 = d+5 = 7 d + 6 d+5 Sec 8. 2 - 17

8. 2 Adding and Subtracting Rational Expressions EXAMPLE 4 Using the Distributive Property When

8. 2 Adding and Subtracting Rational Expressions EXAMPLE 4 Using the Distributive Property When Subtracting Rational Expressions Add or subtract as indicated. The LCD of (e – 2) and (e + 2) is (e – 2)(e + 2). 2 e– 2 – 9 e+2 = 2(e + 2) – (e – 2)(e + 2) 9(e – 2) (e + 2)(e – 2) = 2(e + 2) – 9(e – 2)(e + 2) = 2 e + 4 – 9 e + 18 (e – 2)(e + 2) = – 7 e + 22 (e – 2)(e + 2) Copyright © 2010 Pearson Education, Inc. All rights reserved. Fundamental property Subtract. Distributive property Combine terms in the numerator. Sec 8. 2 - 18

8. 2 Adding and Subtracting Rational Expressions EXAMPLE 5 Adding Rational Expressions with Denominators

8. 2 Adding and Subtracting Rational Expressions EXAMPLE 5 Adding Rational Expressions with Denominators That Are Opposites Add. 4 x– 5 + x 5–x Opposites = 4 x– 5 + x(– 1) (5 – x)(– 1) = 4 x– 5 + –x x– 5 To get a common denominator of x – 5, multiply the second expression by – 1 in both the numerator and the denominator. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8. 2 - 19

8. 2 Adding and Subtracting Rational Expressions EXAMPLE 5 Adding Rational Expressions with Denominators

8. 2 Adding and Subtracting Rational Expressions EXAMPLE 5 Adding Rational Expressions with Denominators That Are Opposites Add. 4 x– 5 + x 5–x Opposites = 4 x– 5 + x(– 1) (5 – x)(– 1) = 4 x– 5 + –x x– 5 = 4–x x– 5 Add the numerators. If we had used 5 – x as the common denominator and rewritten the first expression, we would have obtained x– 4 , 5–x an equivalent answer. Verify this. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8. 2 - 20

8. 2 Adding and Subtracting Rational Expressions EXAMPLE 6 Adding and Subtracting Three Rational

8. 2 Adding and Subtracting Rational Expressions EXAMPLE 6 Adding and Subtracting Three Rational Expressions Add. 3 n+3 + 2 4 + n 2 + 3 n n = 3 n 4(n + 3) 2 + + n(n + 3) Fundamental property The denominator of the second rational expression factors as n(n + 3), which is the LCD for the three rational expressions. = 3 n + 2 + 4(n + 3) n(n + 3) Add the numerators. = 3 n + 2 + 4 n + 12 n(n + 3) Distributive property = = Copyright © 2010 Pearson Education, Inc. All rights reserved. 7 n + 14 n(n + 3) 7(n + 2) n(n + 3) Combine terms. Factor the numerator. Sec 8. 2 - 21

8. 2 Adding and Subtracting Rational Expressions EXAMPLE 7 Add. Subtracting Rational Expressions a–

8. 2 Adding and Subtracting Rational Expressions EXAMPLE 7 Add. Subtracting Rational Expressions a– 2 a+2 – a 2 + 2 a – 3 a 2 – 5 a + 4 = a– 2 a+2 – (a – 1)(a + 3) (a – 1)(a – 4) Factor each denominator. The LCD is (a – 1)(a + 3)(a – 4). = (a – 2)(a – 4) (a + 2)(a + 3) – (a – 1)(a + 3)(a – 4) (a – 1)(a – 4)(a + 3) = (a – 2)(a – 4) – (a + 2)(a + 3) (a – 1)(a + 3)(a – 4) = a 2 – 6 a + 8 – (a 2 + 5 a + 6) (a – 1)(a + 3)(a – 4) Copyright © 2010 Pearson Education, Inc. All rights reserved. Fundamental property Subtract. Multiply in the numerator. Sec 8. 2 - 22

8. 2 Adding and Subtracting Rational Expressions EXAMPLE 7 Add. Subtracting Rational Expressions a–

8. 2 Adding and Subtracting Rational Expressions EXAMPLE 7 Add. Subtracting Rational Expressions a– 2 a+2 – a 2 + 2 a – 3 a 2 – 5 a + 4 = a– 2 a+2 – (a – 1)(a + 3) (a – 1)(a – 4) = (a – 2)(a – 4) (a + 2)(a + 3) – (a – 1)(a + 3)(a – 4) (a – 1)(a – 4)(a + 3) = (a – 2)(a – 4) – (a + 2)(a + 3) (a – 1)(a + 3)(a – 4) = a 2 – 6 a + 8 – (a 2 + 5 a + 6) (a – 1)(a + 3)(a – 4) Multiply in the numerator. = a 2 – 6 a + 8 – a 2 – 5 a – 6) (a – 1)(a + 3)(a – 4) Distributive property = – 11 a + 2 (a – 1)(a + 3)(a – 4) Copyright © 2010 Pearson Education, Inc. All rights reserved. Subtract. Combine terms in the numerator. Sec 8. 2 - 23

8. 2 Adding and Subtracting Rational Expressions EXAMPLE 8 Adding Rational Expressions 2 8

8. 2 Adding and Subtracting Rational Expressions EXAMPLE 8 Adding Rational Expressions 2 8 + b 2 + 4 b + 4 b 2 + 3 a + 2 = 2 8 + (b + 2)2 (b + 1)(b + 2) Factor each denominator. The LCD is (b + 2)2(b + 1) = 2(b + 1) 8(b + 2) + (b + 2)2(b + 1) = 2(b + 1) + 8(b + 2)2(b + 1) = 2 b + 2 + 8 b + 16 (b + 2)2(b + 1) = 10 b + 18 (b + 2)2(b + 1) Copyright © 2010 Pearson Education, Inc. All rights reserved. Fundamental property Add. Distributive property Combine like terms in the numerator. Sec 8. 2 - 24