Digital Lesson Operations on Rational Expressions

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. x2 9 Example: Simplify . x 3 ( x 3)( x 3) x 3 ( x 3)( x 3) , x – 3 0 ( x 3) ( x 3), x 3 Copyright by Houghton Mifflin Company, Inc. All rights reserved. 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 ac b d bd Copyright by Houghton Mifflin Company, Inc. All rights reserved. 3

Example: Multiply 2 2 x 3x x x 2 . 2 2 x 2x 3 x 2x 3 x( x 3) ( x 1)( x 2) ( x 3)( x 1) ( x 3)( x 1) Factor the numerator and denominator of each fraction. x( x 3)( x 1)( x 2) ( x 3)( x 1)( x 3)( x 1) Multiply. 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. 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 a b divisor. The reciprocal of is . b a 2. Multiply the numerators. Then multiply the denominators. 3. Divide by the common factors. 4. Write the answer in simplest form. a c a d ad b d b c bc Copyright by Houghton Mifflin Company, Inc. All rights reserved. 5

2 2 x x y 2 x 2 x y. Example: Divide z z2 x x2 y z2 z 2x 2x2 y 2 x(1 xy) z 2 x(1 xy) z x(1 xy) z 2 2 x(1 xy) z z 2 Copyright by Houghton Mifflin Company, Inc. All rights reserved. Multiply by the reciprocal of the divisor. Factor and multiply. Divide by the common factors. Simplest form 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. x 2x 1 Example: Write the fractions 4x 2 and 6 x 2 12 x in terms of the LCM of the denominators. The LCM of the denominators is 12x2(x – 2). x x 3( x 2) 3( x 2)( x) 2 4x (2 x)( 2 x) 3( x 2) 12 x 2 ( x 2) 2x 1 2 x(2 x 1) 2x 1 2x 2 6 x 12 x 6 x( x 2) 2 x 12 x 2 ( x 2) Copyright by Houghton Mifflin Company, Inc. All rights reserved. LCM 8

To add rational expressions: 1. If necessary, rewrite the fractions with a common denominator. 2. Add the numerators of each fraction. a c a c b b b To subtract rational expressions: 1. If necessary, rewrite the fractions with a common denominator. 2. Subtract the numerators of each fraction. a c a c b b b Copyright by Houghton Mifflin Company, Inc. All rights reserved. 9

2x 5x . Example: Add 14 14 2 x 5 x 7x x 14 14 2 2x 4 2 Example: Subtract 2 . x 4 x 4 2 2( x 2 ) 2x 4 2 x 4 ( x 2)( x 2) ( x 2) Copyright by Houghton Mifflin Company, Inc. All rights reserved. 10

Two rational expressions with different denominators can be added or subtracted after they are rewritten with a common denominator. Example: Add 2x 3 2 6 . x 2x x 4 x 3 6 x( x 2) ( x 2)( x 2) x 3 ( x 2) 6 ( x) x( x 2) ( x 2) ( x 2)( x 2) ( x) ( x 3)( x 2) 6 x x 2 x 6 6 x x( x 2)( x 2) x( x 2)( x 2) ( x 6)( x 1) x 2 5x 6 x( x 2)( x 2) x( x 2)( x 2) Copyright by Houghton Mifflin Company, Inc. All rights reserved. 11

x2 1 2 Example: Subtract 2 . x 1 x 1 x2 1 2 x 1 Add numerators. ( x 1)( x 1) ( x 1)( x 1) Factor. ( x 1)( x 1) ( x 1)( x 1) Divide. 1 Simplest form Copyright by Houghton Mifflin Company, Inc. All rights reserved. 12