Adding and Subtracting Rational Expressions. Add and subtract rational expressions with the same denominator.

Transcription

1 Chapter 7 Section

2 7. Objectives Adding and Subtracting Rational Expressions 1 3 Add and subtract rational expressions with the same denominator. Find a least common denominator. Add and subtract rational expressions with different denominators.

3 Objective 1 Add and subtract rational expressions with the same denominator. Slide 7.- 3

4 Add and subtract rational expressions with the same denominator. 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 least common denominator, and then add or subtract the numerators. Place the result over the common denominator. Step Simplify. Write all answers in lowest terms. Slide 7.- 4

5 EXAMPLE 1 Add or subtract as indicated. 7x y x 3x 7x y x r t r+ t + r t r t r t 6 x x x x x Adding and Subtracting Rational Expressions Same Denominator 6 3x x ( r+ t)( r t) r + t 6 + x x + 3x x x+ 6 x 3 ( )( ) 1 r t 1 x 3 Slide 7.- 5

6 Objective Find a least common denominator. Slide 7.- 6

7 Find a least common denominator. Finding the Least Common Denominator Step 1 Factor each denominator. Step Find the least common denominator. The LCD is the product of all of the different factors from each denominator, with each factor raised to the greatest power that occurs in any denominator. Slide 7.- 7

8 EXAMPLE Find the LCD for each group of denominators. 10 ab,15ab Finding Least Common Denominators Factor. 10a 3 b 5 5 a 3 b 5 15a b a b 6 LCD 3 5 a 3 b 6 30a 3 b 6 z, z + 6 Each denominator is already factored. LCD z(z + 6) Slide 7.- 8

9 EXAMPLE Finding Least Common Denominators (cont d) Find the LCD for each group of denominators. m 16, m + 8m + 16 Factor. m 16 (m +4)(m 4) m + 8m + 16 (m +4) LCD (m +4) (m 4) x x + 1, x 4x + 3, 4x 4 x x + 1 (x 1)(x 1) x 4x + 3 (x 1)(x 3) 4x 4 4(x 1) LCD 4(x 1) (x 3) Slide 7.- 9

10 Objective 3 Add and subtract rational expressions with different denominators. Slide

11 EXAMPLE 3 Add or subtract as indicated. Adding and Subtracting Rational Expressions (Different Denominators) m 4m m 4 4m m 4m m 5 4m 1 y y + 4 ( y + 4) ( + 4) ( ) y ( y + 4) 1 y y y ( y + ) y ( + 4) y( y+ 4) 4 y y y+ 8 y y y ( + 4) y + 8 y y ( + 4) Slide

12 EXAMPLE 4 Subtract. 5x+ 7 x 14 x+ 7 x+ 7 Subtracting Rational Expressions The denominators are already the same for both rational expressions. The subtraction sign must be applied to both terms in the numerator of the second rational expression. 5x + 7 ( x 14) x + 7 5x + 7+ x + 14 x + 7 6x + 1 x ( x + 7) x Slide 7.- 1

13 EXAMPLE 4 Subtract. r + 3 r r 1 Subtracting Rational Expressions (cont d) The LCD is (r )(r 1). ( r ) ( r )( r 1) ( r + 3)( r ) ( r 1)( r ) 1 r + ( r r 6) ( r )( r 1) r r r + ( r )( r 1) 6 r + r+ 4 ( r )( r 1) Slide

14 EXAMPLE 5 Add. Adding and Subtracting Rational Expressions (Denominators Are Opposites) x x To get a common denominator of x 3, multiply both the numerator and denominator of the second expression by 1. 1( 1) ( x)( ) + x x 3 x 3 ( ) + 1 x 3 1 x 3 Slide

15 EXAMPLE 6 Add and subtract as indicated x 5 x x 5x x 5 x x x 5 ( ) ( ) ( x 5) 0 ( x 5) ( x ) 4x 1 + x 5 x x x 5 Adding and Subtracting Three Rational Expressions ( )( x ) x( x 5) 4x x+ x x x x ( 5) x( x 5) x 5 Slide

16 EXAMPLE 7 Subtract. a 4a a a a a a 4a ( a+ 4)( a 1) ( a+ 4)( a+ 3) a+ 4 a 1 a+ 3. LCD is ( )( )( ) Subtracting Rational Expressions a( a+ 3) ( a+ 4)( a 1)( a+ 3) 4a( a 1) ( a+ 4)( a 1)( a+ 3) Slide

17 EXAMPLE 7 Subtracting Rational Expressions (cont d) a( a 3) 4a( a 1) ( a+ 4)( a 1)( a+ 3) + + a 3a 4a 4a ( a+ 4)( a 1)( a+ 3) 5a + a ( a+ 4)( a 1)( a+ 3) Distributive property Combine terms in the numerator. Slide

18 EXAMPLE 8 Add p p p p Adding Rational Expressions ( p 3)( p 3) ( p+ 5)( p 3) LCD is p 3 p+ 5. ( ) ( ) ( p ) ( ) ( ) ( p ) ( )( ) p 3 p+ 5 p+ 5 p 3 Slide

19 EXAMPLE 8 Adding Rational Expressions (cont d) ( p+ ) + ( p ) ( p 3) ( p+ 5) p+ 0 + p 3 ( p 3) ( p+ 5) 5p + 17 ( p 3) ( p+ 5) Distributive property Combine terms in the numerator. Slide