Module 4 Algebraic Fractions In arithmetic, a rational number or fraction is the quotient of two integers where the denominator is not zero. Rational expressions in Algebra play the same role as rational numbers in arithmetic. A rational expression or algebraic fraction is the quotient of two polynomials. Sign of a Fraction A fraction has three signs associated with 1. the sign of the numerator 2. the sign of the denominator and 3. the sign preceding the fraction a a a = = b b b Some Properties used in the Study of Fractions 1. The value of a fraction is not changed when both the numerator and the denominator are multiplied or divided by the same number which is not zero. a ac = ,c0 b bc a a/c = , c 0 b b/c 2. Given two fractions a b b c a b a+b = ,c0 c c c 3. Given the fractions a c b d a c ac = , bd 0 b d bd a c a d ad = = , bc 0 b d b c bc Reduction of Fractions to Lowest Terms To reduce fractions to lowest terms, divide both numerator and denominator by their greatest common factor ( G.C.F.). Examples: 3 2 1. 12 x y 4 x y 3 x = 2 = 3 2 4 16 x y 4 x y 4 y 2. y 2 10 y+25 2 y 7 y+10 = 3. 3 m3 n nm 3 ( mn ) =3 ( mn ) = ( y5 )2 y5 = ( y 2 )( y 5 ) y2 Addition and Subtraction of Fractions The sum of two or more fractions having the same denominator is a fraction whose numerator is the sum of the numerators and whose denominator is the common denominator. When fractions have different denominators, they must be changed to equivalent fractions having a common denominator. A polynomial that is exactly divisible by two or more polynomials is a common multiple of the polynomials. The least common denominator ( LCD ) of two or more fractions is the least common multiple ( LCM ) of the denominators of the fractions. To find the LCD of two or more fractions, factor each denominator. The LCD is the product of all the different prime factors with each factor raised to the largest exponent with which the factor occurs in any denominator. Example: Find the LCD of 1 1 1 , 2 , x +5 x 6 x 9 x3 2 x2 + 5x + 6 = (x+2)(x+3) x2 - 9 = (x+3)(x-3) x-3 = (x-3) LCD = ( x + 2 ) ( x + 3 ) ( x - 3 ) To add two or more fractions, find the LCD of the fractions, reduce the fractions to equivalent fractions having the LCD for their denominator and the combine the numerators, writing the sum over the LCD. Examples: 2 g. x x 3 + 2 2 2 x 6 x9 x 5 x+ 6 x 9 2 {LCD = ( x3 ) ( x +3 )( x2 ) } 2 = = x x 3 + 2 ( x3 ) ( x3 ) ( x2 ) ( x +3 ) ( x3 ) { Express denominators in LCD form} x ( x+3 )( x2 ) + x 2 ( x 3 ) ( x +3 ) +3 ( x3 )( x2 ) ( x3 )2 ( x +3 ) ( x2 ) {Use the Distributive Law } 3 = 4 = 2 4 2 2 x + x 6 x + x 9 x 3 x +15 x 18 ( x3 )2 ( x +3 ) ( x2 ) 3 { Add the numerators} 2 x + x 11 x +9 x18 ( x3 )2 ( x+ 3 ) ( x2 ) Note: A common error in sign is when a fraction with several terms in the numerator is subtracted. Remember to distribute the minus sign to every term of the numerator of the subtrahend. Multiplication of Fractions To multiple fractions, factor completely all numerators and denominators. Cancel all common factors. Multiply the remaining factors in the numerators and the remaining factors in the denominators. Examples: e. 2 2 10 x 2 +10 y 2 2 x +2 y 10 ( x + y ) 2 ( x+ y ) 2 = 2 ( x+ y ) 2 ( x + y ) 5 ( x2 + y2 ) 5 x +5 y 2 =2 x 5x + f. x 4x 5 6 x x - 6 x 6x + 5 x 5x + x 4x 6 5 x 6x + x x 6 5 Reciprocal of a Rational Expression = (x + 1)(x 5) (x 3)(x 2) (x + 2)(x 3) (x 1)(x 5) = x+1 x2 x+2 x1 = x x 2 x + x 2 Two rational number a c b d are reciprocals if their product is 1. Similarly, two rational expressions or algebraic fractions are reciprocals if their product is 1. Division of Fractions To divide fractions, multiply the dividend by the reciprocal of the divisor. Examples: 2 x5 y 5 y 2 x 2 x5 y c +1 = 5 c+5 c +1 5 c +5 5 y2 x e. = 2 x5 y c+ 1 5 ( c+ 1 ) ( 2 x5 y ) = 1 5 Complex Fractions A complex fraction has one or more fractions in its numerator or denominator or both. Steps in performing and simplifying complex fraction : 1. Factor everything. 2. Determine the total least common denominator, using the maximum power of each factor. 3. Multiply all terms by the LCD. 4. Combine like terms. 5. Factor and Cancel. Note: Usually you will not have to do all of the steps. Examples: 1 x 2 3+ 2 x 4+ a. b. 7 7 1- 1(x + 1) x+1 (x + 1) x+1 = 4 4 + 1 (x + 1) + 1(x + 1) x+1 x+1 x+1-7 = 4+x+1 c. x-6 = x+5 ( x+ 2 )+ ( x+1 ) 1 1 + ( x+1 ) ( x+ 2 ) x +1 x +2 = 1 1 ( x+2 )( x+ 1 ) x +1 x +2 ( x+1 ) ( x+ 2 ) = = 2 x +3 ( x+ 1 )( x +2 ) 1 ( x+ 1 )( x +2 ) ( x+1 ) ( x+ 2 ) 2 x+3 1 ( x +1 ) ( x+2 ) = 2x + 3 Assignment 4 A. Reduce the following fractions to lowest terms: 1. 18 x y 2 z 36 x 2 y z 2 2. 4 mn 3 2 64 m n 3. ay+ 5 y a2 +5 a 4. 12 xy +6 xyz 2 2 3 x y+ 9 x yz 2 5. y +6 y +9 2 y + y6 6. 3 c 29 cd54 d2 3 c 26 cd72 d 2 7. 3 p+5 q 2 2 25 q 9 p 8. 5 y 245 y +90 1805 y 2 9. a3a 2 b+ a b2b3 3 2 2 3 a +a b+ a b +b 10. m3 +n3 + m+n m2mn+ n2+ 1 B. Combine into a single fraction expressing each answer in lowest terms. 11. x+ 5 2 x5 12. 2 x 2x +2 x1 x 1 13. y +1 y1 - 1 2y 3y 14. x 1 x3 - 2 x +3 x+ 2 2 x +3 x2 2 x +3 1 2x + 2 x4 x x 16 15. 16. 6m 3 + m 36 6m 17. 10 25 x y+ 2 y x2 18. 7 5x 2x x3 x 29 x 2 +6 x+ 9 19. 12 8 x+ 4 + 2 3 x + x+ 4 3 x 2 x8 20. 2 2 2 a+1 a a1 + 2 2 2 2 a - a1 a + a2 2 a +5 a+2 C. Perform the indicated operations expressing each result in lowest terms. 21. 2 x 2 y 3 ax 4 x y 2 6 a2 z 4 22. 16 x 2 39 y 13 y 2 64 x 23. 10 ab 4 15 b2 2 25 b3 a b 24. a+2 b a3 8 b3 a2 b a3 + 8 b3 25. 1x x2 1 1 y 2 1 y 2 26. 3 x12 ( x4 ) 5x 27. 7 y +21 x y 26 xy +9 x 2 y 29 x 2 y 22 xy3 x 2 28. 9 y 16 x 2 xy 2 y 2 2 2 2 x 2 xy 3 y xy4 x 2 2 2 29. x 2 + x2 x 2 +2 x x 27 x 5 x 238 x +21 30. p3 p3 27 p 22 p3 p +3 p2 3 p+ 9 p2 9 D. Simplify the following: 31. 64 1 10+ 2 4 32. 33. 1 x x 4 x y y x x y + +7 y x 34. 6 x1 2 y y1 +3 y 35. 1 6 + 3 x+ 2 3 x2 1 4 9 2 x 36. x y y x 2x y + y 2x 37. 4 3 1 + a b c 1 5 2 + ab bc ac 38. a a + x+ y x y a a 2 2 x + y x +y 2 x 39. 2 1 1+ 40. 1 x y1 y +1 y +1 y1 1 1 + y +1 y1 - End