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# 11.4

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### 11.4

1. 1. Adding and subtracting Rational Expressions with the Same Denominator Let a, b, and c be polynomials where Algebra: a b a+b + = c c c c ¹ 0. a b a-b - = c c c 2 3 2+3 5 Examples: = + = x x x x 5 2 5- 2 3 = = 4x 4x 4x 4x
2. 2. Example 1 Add and subtract with the same denominator 5 7 12 a. + = 3x 3x 3x Add numerators. 3•4 = 3•x Factor and divide out common factor. 4 = x Simplify. x +5 3x – ( x + 5 ) – b. = x – 1 x – 1 x –1 3x 2x – 5 = x – 1 Subtract numerators. Simplify.
3. 3. Least Common Denominator The least common denominator (LCD) of two or more rational expressions is the same as the least common multiple of the denominators of the rational expressions.
4. 4. Example 2 Find the LCD of rational expressions Find the LCD of the rational expressions. 1 r +3 , a. 4r 10r 2 5 3x + 4 , b. ( x – 3 )2 x2 – x – 6 3 , c + 8 c. c – 2 2c + 7
5. 5. Example 2 Find the LCD of rational expressions SOLUTION a. Find the least common multiple ( LCM ) of 4r and 10r 2. 4r = 22 • r 10r2 = 2 • 5 • r 2 LCM = 22 • 5 • r 2 = 20r 2 ANSWER 1 r +3 The LCD of and is 20r2. 10r2 4r
6. 6. Example 2 Find the LCD of rational expressions b. Find the least common multiple ( LCM ) of ( x – 3 )2 and x 2 – x – 6. ( x – 3 )2 = ( x – 3 )2 x2 – x – 6 = ( x – 3 ) • ( x + 2 ) LCM = ( x – 3 )2 ( x + 2 ) ANSWER 3x + 4 5 The LCD of and is ( x – 3 )2 ( x + 2). ( x – 3 )2 x2 – x – 6
7. 7. Example 2 Find the LCD of rational expressions c. Find the least common multiple of c – 2 and 2c + 7. Because c – 2 and 2c + 7 cannot be factored, they don’t have any factors in common. The least common multiple is their product, ( c – 2 ) ( 2c + 7 ). ANSWER 3 c + 8 The LCD of and is (c – 2)( 2c + 7). c– 2 2c + 7
8. 8. Different Denominators To add or subtract rational expressions that have different denominators, use the LCD to write equivalent rational expressions that have the same denominator (just as you would for numerical fractions). In the example below, 12x 2 is the LCD.
9. 9. Example 3 Add expressions with different denominators Find the sum 9 5 . + 8x 2 12x 3 9 • 3x 5 • 2 9 5 = + + 2 3 2 • 3x 8x 12x 8x 12x 3 • 2 Rewrite fractions using LCD, 24x 3. 27x 10 = + 3 24x 24x 3 Simplify numerators and denominators. 27x + 10 = 24x 3 Add fractions.
10. 10. Example 4 Subtract expressions with different denominators Find the difference 10 7x . – 3x x + 2 10 7x 10 ( x + 2 ) 7x ( 3x ) Rewrite fractions – – = ( x + 2 ) ( 3x ) using LCD, 3x ( x + 2 ). 3x ( x + 2 ) 3x x + 2 10 ( x + 2 ) – 7x ( 3x ) = 3x ( x + 2 ) = –21x 2 + 10x + 20 3x ( x + 2 ) Subtract fractions. Simplify numerator.
11. 11. Example 5 Subtract expressions with different denominators x + 4 x – 1 – Find the difference 2 . 2 + 2x – 8 x + 3x – 10 x x + 4 x – 1 – 2 + 3x – 10 x x2 + 2x – 8 x + 4 x – 1 – Factor denominators. = ( x – 2 ) ( x + 5 ) ( x + 4 )( x – 2 ) ( x + 4 )( x + 4 ) ( x – 1 )( x + 5 ) – = ( x + 4 )( x – 2 )( x + 5 ) ( x – 2 )( x + 5 )( x + 4 ) Rewrite fractions using LCD, ( x – 2 ) ( x + 5 ) ( x + 4 ).