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Digital Lesson Operations on Rational Expressions Rational expressions are fractions in which the numerator and denominator are polynomials and the denominator does not equal zero. x 9 2 Example: Simplify x 3 . ( x 3)( x 3) x 3 ( x 3)( x 3) , ( x 3) ( x 3) , Copyright © by Houghton Mifflin Company, Inc. All rights reserved. x–30 x3 2 To multiply rational expressions: 1. Factor the numerator and denominator of each fraction. 2. Multiply the numerators and denominators of each fraction. 3. Divide by the common factors. 4. Write the answer in simplest form. a c • b d Copyright © by Houghton Mifflin Company, Inc. All rights reserved. ac bd 3 x 3x 2 Example: Multiply x( x 3) ( x 3)( x 1) • x 2x 3 2 x x2 2 • x 2x 3 2 . ( x 1)( x 2) Factor the numerator and ( x 3)( x 1) denominator of each fraction. x( x 3)( x 1)( x 2) ( x 3)( x 1)( x 3)( x 1) x( x 3)( x 1)( x 2) ( x 3)( x 1)( x 3)( x 1) x( x 2) ( x 1)( x 3) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Multiply. Divide by the common factors. Write the answer in simplest form. 4 To divide rational expressions: 1. Multiply the dividend by the reciprocal of the divisor. The reciprocal of a is b b . a 2. Multiply the numerators. Then multiply the denominators. 3. Divide by the common factors. 4. Write the answer in simplest form. a b c d Copyright © by Houghton Mifflin Company, Inc. All rights reserved. a d • b c ad bc 5 xx y 2 Example: Divide 2x 2x y 2 z xx y z 2 z z • 2 Factor and multiply. 2 2 x(1 xy ) • z z Multiply by the reciprocal of the divisor. 2 2 x(1 xy ) • z x(1 xy ) • z 2 2 2x 2x y x(1 xy ) • z . Divide by the common factors. Simplest form 2 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 The least common multiple (LCM) of two or more numbers is the least number that contains the prime factorization of each number. Examples: 1. Find the LCM of 10 and 4. 10 = (5 • 2) factors of 10 4 = (2 • 2) LCM = 2 • 2 • 5 = 20 factors of 4 2. Find the LCM of 4x2 + 4x and x2 + 2x + 1. 4x2 + 4x = (4x)(x +1) = 2 • 2 x (x + 1) x2 + 2x + 1 = (x +1)(x +1) factors of x2 + 2x + 1 LCM = 2 • 2 x (x +1)(x +1) = 4x3 + 8x2 + 4x factors of 4x2 + 4x Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 Fractions can be expressed in terms of the least common multiple of their denominators. Example: Write the fractions x 4x 2 and 2x 1 6 x 12 x 2 in terms of the LCM of the denominators. The LCM of the denominators is 12x2(x – 2). x 4x 2 x ( 2 x)(2 x) 2x 1 6 x 12 x 2 • 3( x 2) 3( x 2) 2x 1 6 x( x 2) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. • 2x 2x 3( x 2)( x) 12 x ( x 2) 2 2 x( 2 x 1) LCM 12 x ( x 2) 2 8 To add rational expressions: 1. If necessary, rewrite the fractions with a common denominator. 2. Add the numerators of each fraction. a b c b ac b To subtract rational expressions: 1. If necessary, rewrite the fractions with a common denominator. 2. Subtract the numerators of each fraction. a b Copyright © by Houghton Mifflin Company, Inc. All rights reserved. c b ac b 9 Example: Add 2x 14 5x . 14 2 x 5x 14 Example: Subtract x 4 2 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14 2x 7x 2x 4 x 4 2 x 2 4 x 4 2 . 2( x 2) ( x 2)( x 2) 2 ( x 2) 10 Two rational expressions with different denominators can be added or subtracted after they are rewritten with a common denominator. x 3 Example: Add x 2x 2 6 x 3 x( x 2) x 3 x( x 2) . x 4 2 • 6 ( x 2)( x 2) ( x 2) ( x 2) ( x 3)( x 2) 6 x x( x 2)( x 2) x 5x 6 2 x( x 2)( x 2) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 • ( x) ( x 2)( x 2) ( x) x x 6 6x 2 x( x 2)( x 2) ( x 6)( x 1) x( x 2)( x 2) 11 Example: Subtract x 2 x 1 2 1 x 1 2 x 1 . 2 Add numerators. x 1 2 ( x 1)( x 1) ( x 1)( x 1) ( x 1)( x 1) ( x 1)( x 1) 1 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Factor. Divide. Simplest form 12