6.1. Rational Expressions and Functions; Multiplying and Dividing. Copyright 2016, 2012, 2008 Pearson Education, Inc. 1

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1 6.1 Rational Expressions and Functions; Multiplying and Dividing 1. Define rational expressions.. Define rational functions and give their domains. 3. Write rational expressions in lowest terms. 4. Multiply rational expressions. 5. Find reciprocals of rational expressions. 6. Divide rational expressions. Copyright 016, 01, 008 Pearson Education, Inc. 1

2 Objective 1 Define rational expressions. Copyright 016, 01, 008 Pearson Education, Inc.

3 Define rational expressions. In arithmetic, a rational number is the quotient of two integers, with the denominator not 0. In algebra, a rational expression, or algebraic fraction, is the quotient of two polynomials, again with the denominator not 0. x , a, m +, x x +, and or x x y 4 m 4x + 5x 1 Rational expressions are elements of the set P P and Q are polynomials, where Q 0. Q Copyright 016, 01, 008 Pearson Education, Inc. 3

4 Objective Define rational functions and give their domains. Copyright 016, 01, 008 Pearson Education, Inc. 4

5 Define rational functions and give their domains. A rational function is defined by a quotient of polynomials and has the form Px ( ) f( x) =, where Qx ( ) 0. Qx ( ) The domain of a rational function includes all real numbers except those that make Q(x) that is, the denominator equal to 0. Copyright 016, 01, 008 Pearson Education, Inc. 5

6 Classroom Example 1 Finding Domains of Rational Functions Give the domain of each rational function using setbuilder notation and interval notation. a. x + 6 f( x) = x x 6 Values of x that make the denominator 0 must be excluded from the domain. To find these values, set the denominator equal to 0 and solve the resulting equation. x x 6= 0 x x is a real number, x,3 ( x+ )( x 3) = 0 x+ = 0 or x 3 = 0 x= or x= 3 { } (, ) (, 3) ( 3, ) Copyright 016, 01, 008 Pearson Education, Inc. 6

7 Classroom Example 1 Finding Domains of Rational Functions (cont.) Give the domain of each rational function using setbuilder notation and interval notation. b. 3+ x f( x) = 5 The denominator, 5, cannot ever be 0, so the domain includes all real numbers (, ). Copyright 016, 01, 008 Pearson Education, Inc. 7

8 Objective 3 Write rational expressions in lowest terms. Copyright 016, 01, 008 Pearson Education, Inc. 8

9 Fundamental Property of Rational Numbers Copyright 016, 01, 008 Pearson Education, Inc. 9

10 Writing a Rational Expression in Lowest Terms Step 1 Factor both numerator and denominator to find their greatest common factor (GCF). Step Apply the fundamental property. Divide out common factors. Copyright 016, 01, 008 Pearson Education, Inc. 10

11 Classroom Example Write each rational expression in lowest terms. a. b. y y = + y 3 3y+ ( y+ 3)( y 1) ( y )( y 1) = y + y y + y Writing Rational Expressions in Lowest Terms 1+ p c. 1 + p The denominator cannot be factored, so this expression cannot be simplified further and is in lowest terms. = Copyright 016, 01, 008 Pearson Education, Inc (1 p)(1 p p ) p = 1 p+ p

12 Classroom Example 3 Write each rational expression in lowest terms. a. r 1 1 r = r 1 ( 1)( r 1) 1 = = 1 1 Writing Rational Expressions in Lowest Terms p b. 4 p p = 1( p 4) = p 1( p )( p+ ) 1 1 = = ( 1)( p + ) p + Copyright 016, 01, 008 Pearson Education, Inc. 1

13 Quotient of Opposites In general, if the numerator and the denominator of a rational expression are opposites, then the expression equals 1. Based on this result, the following are true statements. However, the following expression cannot be simplified further. Copyright 016, 01, 008 Pearson Education, Inc. 13

14 Objective 4 Multiply rational expressions. Copyright 016, 01, 008 Pearson Education, Inc. 14

15 Multiply Rational Expressions Step 1 Factor all numerators and denominators as completely as possible. Step Apply the fundamental property. Step 3 Multiply remaining factors in the numerators and remaining factors in the denominators. Leave the denominator in factored form. Step 4 Check to be sure the product is in lowest terms. Copyright 016, 01, 008 Pearson Education, Inc. 15

16 Classroom Example 4 Multiply. a. c + c c 4c+ 4 c 4 c c Multiplying Rational Expressions cc ( + ) ( c )( c ) = ( c+ )( c ) cc ( 1) = c c 1 b. m m m+ 4 ( m+ 4)( m 4) 1 = m+ m+ 4 = m m + 4 Copyright 016, 01, 008 Pearson Education, Inc. 16

17 Objective 5 Find reciprocals of rational expressions. Copyright 016, 01, 008 Pearson Education, Inc. 17

18 Finding the Reciprocal When dividing rational numbers, we use reciprocals. The reciprocal of a rational expression is defined in the same way. Two rational expressions are reciprocals of each other if they have a product of 1. Finding the Reciprocal To find the reciprocal of a nonzero rational expression, interchange the numerator and denominator of the expression. Copyright 016, 01, 008 Pearson Education, Inc. 18

19 Objective 6 Divide rational expressions. Copyright 016, 01, 008 Pearson Education, Inc. 19

20 Dividing Rational Expressions To divide two rational expressions, multiply the first expression (the dividend) by the reciprocal of the second expression (the divisor). Copyright 016, 01, 008 Pearson Education, Inc. 0

21 Classroom Example 5 Divide. Dividing Rational Expressions 5 p + 15p + 6 q + q 4 q a. b q 3q 6 5p + 5 q + q 3q 6 = = 6 15 p q 4 q qq ( + ) 3( q ) 5p + 5 = = 5 + q ( q)( + q) 6 3(5 p + ) qq ( + ) 3( q ) = q ( 1)( q )( q+ ) = 18 = 3q 5 + q Copyright 016, 01, 008 Pearson Education, Inc. 1