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REVIEW OF ADDING & SUBTRACTING FRACTIONS 19 5 21 18 You cannot add or subtract fractions unless they have a common denominator. The LEAST COMMON DENOMINATOR is the smallest number that contains all the prime factors of both numbers. 1) Write the prime factorization of each denominator. 21 = 3∙7 21 18 = 2∙3 ∙3 3 7 18 9 2 3 3 2) Circle all the prime factors of the first denominator. 21 = 3 ∙ 7 18 = 2 ∙ 3 ∙ 3 Go to the next denominator’s prime factorization and scratch out factors that are in the first denominator. Then circle all the prime factors of the second denominator that DON’T appear in the first denominator's prime factors. Notice that 21 = 3∙7, which has a 3 in the prime factorization. The second denominator is 18 = 2 ∙3 ∙3, which has two 3’s. One 3 already appeared in the prime factorization of 21. The LCD is the circled factors. LCD = 3∙7∙2 3=126 3) Rewrite each fraction into an equivalent fraction with the LCD as the denominator. 19 6 114 21 6 126 5 7 35 18 7 126 4) Now perform the subtraction by subtracting the numerators and leaving the denominator the same. 19 5 114 35 79 21 18 126 126 126 Example Subtract: 2 5 y 18 Are the denominators, y and 18, the same? Then, find the LCD of y and 18. But we don’t know what y is, so how could we know if it goes into 18? When you have denominators where one is a variable term and the other is a constant term, find the Least Common Multiple of the coefficient and the constant , and then find the Least Common Multiple of the variables. The coefficient of y is 1. The other denominator is just 18. What is the LCM of 1 and 18? Remember, LCM means what’s the smallest number that both 1 and 18 will go into? LCM = 18 ____ However, the Least Common Denominator will have to include the the LCM of the variables. Since the other denominator doesn’t have a variable, the LCD of the variables is just y. So the LCD is 18y_ Now rewrite each fraction so that the denominator is ____ 2 18 36 y 18 18y 5 y 5y 18 y 18y The new expression which now has like terms is: Example Subtract: 2 5 3y 6 y2 Are the denominators, 3y and 6y2, the same? Then, find the LCD of 3y and 6y2. The coefficient of 3y is 3. The coefficient of 6y2 is 6. What is the LCM of 3 and 6? Remember, LCM means what’s the smallest number that both 3 and 6 will go into? LCM = __6__ However, the Least Common Denominator will have to include the the LCM of the variables. What’s the LCM of y and y2? _y2 So the LCD is _6y2____ Now rewrite each fraction so that the denominator is ____ The new expression which now has like terms is: 2 3y 5 6 Add : x-3 6 2 2 x -2 x x 4 x-3 6 x( x 2) ( x 2)( x 2) Factor each denominator before finding the LCD. x2-2x = x(x-2) x2-4 = (x-2)(x+2) Like doing prime factorization, we circle the factors in the first denominator, and then circle the factors in the second denominator that are NOT in the first denominator. LCD = x(x-2)(x+2) Now change each fraction to have the LCD for its denominator. Add : x-3 x(x-2 ) (x+2) (x+2) 6 ( x 2)( x 2) x x ( x 3)( x 2) 6 x x 2 3x 2 x 6 6 x x 2 5x 6 x( x 2)( x 2) x( x 2)( x 2) x( x 2)( x 2) ( x 6)( x 1) x( x 2)( x 2) Subtract: 2x 5 x 3 3 x Before trying to find an LCD, notice that 3 - x = -(x - 3) So the expression can be rewritten as: 2x 5 x 3 ( x 3) or 2x 5 x 3 x 3 REMEMBER, when additing fractions, once you have a least common denominator, you ONLY add the numerators and you leave the denominator the same. 2x 5 x3 COMPLEX FRACTIONS COMPLEX FRACTIONS are just fractions inside fractions. They are easily simplified by just multiplying the numerator and denominator of the “main” fraction by the LCD of all the fractions. Example 1: Simplify : 1 1 x 2 1 1 2 x 4 The denominators in this complex fraction are: x, 2, x2 and 4. LCD = 4x2 1 1 x 2 1 1 2 4 x 4x2 4x2 4x2 4x2 2 4 x 2 x 2 x(2 x) 2x x 2 22 2 4 x 4x 4x (2 x)( 2 x) (2 x) 2 4 Yeah! No more complex fractions! x Now factor this rational expression in order to cancel common factors and reduce it.