Rational Expressions and Functions Chapter 8 Rational Expressions and Functions

Adding and Subtracting Rational Expressions 8.2 Adding and Subtracting Rational Expressions

8.2 Adding and Subtracting Rational Expressions Objectives Add and subtract rational expressions with the same denominator. Find a least common denominator. Add and subtract rational expressions with different denominators. Copyright © 2010 Pearson Education, Inc. All rights reserved.

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.

8.2 Adding and Subtracting Rational Expressions EXAMPLE 1 Adding and Subtracting Rational Expressions with the Same Denominator Add or subtract as indicated. 4m 7 5n + 4m + 5n 7 = Add the numerators. (a) Keep the common denominator. 1 g3 5 – 1 – 5 g3 = Subtract the numerators; keep the common denominator. (b) 4 g3 = – Simplify. Copyright © 2010 Pearson Education, Inc. All rights reserved.

8.2 Adding and Subtracting Rational Expressions EXAMPLE 1 Adding and Subtracting Rational Expressions with the Same Denominator Add or subtract as indicated. a a2 – b2 b – a – b a2 – b2 = Subtract the numerators; keep the common denominator. (c) a – b (a – b)(a + b) = Factor. 1 a + b = Lowest terms Copyright © 2010 Pearson Education, Inc. All rights reserved.

8.2 Adding and Subtracting Rational Expressions EXAMPLE 1 Adding and Subtracting Rational Expressions with the Same Denominator Add or subtract as indicated. 5 k2 + 2k – 15 + k (d) = 5 + k k2 + 2k – 15 Add. = 5 + k (k – 3)(k + 5) Factor. = 1 k – 3 Lowest terms Copyright © 2010 Pearson Education, Inc. All rights reserved.

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.

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) 4m3n2, 6m2n5 Factor each denominator. 22 · m3 · n2 2 · 3 · m2 · n5 Choose the factors with the greatest exponents. LCD = 22 · 3 · m3 · n5 = 12m3n5 Copyright © 2010 Pearson Education, Inc. All rights reserved.

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.

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) n2 – 3n – 10, n2 – 8n + 15 Factor the denominators. n2 – 3n – 10 = (n – 5)(n + 2) n2 – 8n + 15 = (n – 5)(n – 3) The LCD, divisible by both polynomials, is (n – 5)(n + 2)(n – 3). Factor. Copyright © 2010 Pearson Education, Inc. All rights reserved.

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) 4h2 – 12h, 3h – 9 4h2 – 12h = 4h(h – 3) 3h – 9 = 3(h – 3) The LCD is 4h·3·(h – 3) = 12h(h – 3). Factor. Copyright © 2010 Pearson Education, Inc. All rights reserved.

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) g2 – 2g + 1, g2 + 3g – 4, 5g + 20 g2 – 2g + 1 = (g – 1)2 g2 + 3g – 4 = (g – 1)(g + 4) 5g + 20 = 5(g + 4) The LCD is 5(g – 1)2(g + 4). Factor. Copyright © 2010 Pearson Education, Inc. All rights reserved.

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 3z and 9z is 9z. 3z 7 9z 1 + (a) 3z 7 9z 1 + = 3z · 3 7 · 3 Fundamental property = 9z 21 1 + = 9z 21 + 1 Add the numerators. = 9z 22 Copyright © 2010 Pearson Education, Inc. All rights reserved.

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). y 2 y – 4 3 – = y(y – 4) 2(y – 4) y · 3 – (b) Fundamental property = y(y – 4) 2y – 8 3y – Distributive and commutative properties = y(y – 4) 2y – 8 – 3y Subtract the numerators. = y(y – 4) –y – 8 Combine like terms in the numerator. Copyright © 2010 Pearson Education, Inc. All rights reserved.

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. Subtract the numerators; keep the common denominator. d + 5 d – 6 8d – = d + 5 8d – (d – 6) = d + 5 8d – d + 6 = d + 5 7d + 6 Combine terms in the numerator. Copyright © 2010 Pearson Education, Inc. All rights reserved.

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). e + 2 9 e – 2 2 – = (e – 2)(e + 2) 2(e + 2) (e + 2)(e – 2) 9(e – 2) – Fundamental property = (e – 2)(e + 2) 2(e + 2) – 9(e – 2) Subtract. = (e – 2)(e + 2) 2e + 4 – 9e + 18 Distributive property = (e – 2)(e + 2) –7e + 22 Combine terms in the numerator. Copyright © 2010 Pearson Education, Inc. All rights reserved.

8.2 Adding and Subtracting Rational Expressions EXAMPLE 5 Adding Rational Expressions with Denominators That Are Opposites 5 – x x x – 5 4 + = (5 – x)(–1) x(–1) x – 5 4 + Add. = x – 5 –x 4 + Opposites 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.

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

8.2 Adding and Subtracting Rational Expressions EXAMPLE 6 Adding and Subtracting Three Rational Expressions n2 + 3n 2 n + 3 3 + n 4 = n(n + 3) 2 3n + 4(n + 3) Add. Fundamental property The denominator of the second rational expression factors as n(n + 3), which is the LCD for the three rational expressions. = n(n + 3) 3n + 2 + 4(n + 3) Add the numerators. = n(n + 3) 3n + 2 + 4n + 12 Distributive property = n(n + 3) 7n + 14 Combine terms. = n(n + 3) 7(n + 2) Factor the numerator. Copyright © 2010 Pearson Education, Inc. All rights reserved.

8.2 Adding and Subtracting Rational Expressions EXAMPLE 7 Subtracting Rational Expressions Add. a2 + 2a – 3 a – 2 – a2 – 5a + 4 a + 2 = (a – 1)(a + 3) a – 2 – (a – 1)(a – 4) a + 2 Factor each denominator. The LCD is (a – 1)(a + 3)(a – 4). = (a – 1)(a + 3)(a – 4) (a – 2)(a – 4) – (a – 1)(a – 4)(a + 3) (a + 2)(a + 3) Fundamental property = (a – 1)(a + 3)(a – 4) (a – 2)(a – 4) – (a + 2)(a + 3) Subtract. = (a – 1)(a + 3)(a – 4) a2 – 6a + 8 – (a2 + 5a + 6) Multiply in the numerator. Copyright © 2010 Pearson Education, Inc. All rights reserved.

8.2 Adding and Subtracting Rational Expressions EXAMPLE 7 Subtracting Rational Expressions Add. a2 + 2a – 3 a – 2 – a2 – 5a + 4 a + 2 = (a – 1)(a + 3) a – 2 – (a – 1)(a – 4) a + 2 = (a – 1)(a + 3)(a – 4) (a – 2)(a – 4) – (a – 1)(a – 4)(a + 3) (a + 2)(a + 3) = (a – 1)(a + 3)(a – 4) (a – 2)(a – 4) – (a + 2)(a + 3) Subtract. = (a – 1)(a + 3)(a – 4) a2 – 6a + 8 – (a2 + 5a + 6) Multiply in the numerator. = (a – 1)(a + 3)(a – 4) a2 – 6a + 8 – a2 – 5a – 6) Distributive property = (a – 1)(a + 3)(a – 4) –11a + 2 Combine terms in the numerator. Copyright © 2010 Pearson Education, Inc. All rights reserved.

8.2 Adding and Subtracting Rational Expressions EXAMPLE 8 Adding Rational Expressions Add. b2 + 4b + 4 2 + b2 + 3a + 2 8 = (b + 2)2 2 + (b + 1)(b + 2) 8 Factor each denominator. The LCD is (b + 2)2(b + 1) = (b + 2)2(b + 1) 2(b + 1) + 8(b + 2) Fundamental property = (b + 2)2(b + 1) 2(b + 1) + 8(b + 2) Add. = (b + 2)2(b + 1) 2b + 2 + 8b + 16 Distributive property = (b + 2)2(b + 1) 10b + 18 Combine like terms in the numerator. Copyright © 2010 Pearson Education, Inc. All rights reserved.