HFCC Math Lab Intermediate Algebra - 7 FINDING THE LOWEST COMMON DENOMINATOR (LCD)

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1 HFCC Math Lab Intermediate Algebra - 7 FINDING THE LOWEST COMMON DENOMINATOR (LCD) Adding or subtracting two rational expressions require the rational expressions to have the same denominator. Example : Adding two rational expressions with the common denominator x+ 5 3 x x = 5 3 x 8 = x When the denominators of two rational expressions are not the same we write equivalent rational expression for both or one of the rational expressions so that they both have same denominator. To find the equivalent rational expressions we multiply the numerator and denominator of a rational function with same non-zero factor. Example : A list of equivalent rational expressions to x x 0 0(x 0 0( x x x x(x x( assuming x 0 x (x 3) (x 3) (x ( x 3) ( ( x 3) assuming x 3 0 x 0 x (3x 5) 0 x (3x 5) 0 x (3x 5)(x 0 (3 5)(5 x x x assuming 0 x (3x 5) 0 Remark: Try to reason out why every rational expression in the list above is equivalent to x We write equivalent rational expressions so that the two rational expressions have common denominator the natural choice is the product of the two denominators. Remark: Try to reason why the above statement makes sense. (Hint: You may have to use your knowledge on equivalent rational expressions to do so). Revised /09

2 Example 3: Some of the multiples of the polynomial 6 x( are 6 x(x 6 x(x 3 6 x(x 8 x(x x 6 x(x 6 x (x 5 6 5(x 6 x(x 30 x(x 3 7 x (x 6 x(x 4 x (x Remark: Try to reason out why every expression in the above list is a multiple of 6 x( The best choice for the denominator is the Least Common Multiple (LCM) of the denominators of two expressions. We also refer to this value as the Least Common Denominator (LCD) Remark: Try to reason why the above statement makes sense. (Hint: Least means smallest multiple of a polynomial. A smallest multiple of a polynomial has the least degree and yet can be written as a product of both the polynomials under consideration.) To find the least common denominator (LCD) of two or more fractional expression you can follow this procedure:. Factor each denominator completely ; i.e. as bases with exponents.. Write the product of all the different 3. Raise each base to the highest exponent to which it is raised in any single denominator Remark: A base and its opposite (negative) can be considered the same when find the LCD; any (- factor can be attached to the numerator. Example 4: Simplifying rational expression with a - factor in the denominator ( x ) x Example 5: Find the Least Common Denominator (LCD) of the expressions x x 4 x Step : Factor each denominator completely Step : Write the product of all the different Step 3: Raise each base to the highest exponent Revised /09

3 y 7 Example 6: Find the Least Common Denominator (LCD) of the expressions 3 8x y 0xy z 5yz 8x y 0xy z 5yz 3 x y y 35 yz 3 z Step : Factor each denominator completely 3 5 x y z Step : Write the product of all the different x y z x y z Step 3: Raise each base to the highest exponent Example 7: Find the Least Common Denominator (LCD) of the expressions a a ( a a a a ( ( a ( a ( a ( a Step : Factor each denominator completely ( a ( a Step : Write the product of all the different ( a ( a Step 3: Raise each base to the highest exponent Example 8: Find the LCD of the expressions 3 8x 7 x 6x 9 x 3x 9 3 8x 7 (x 3) (4 x 6x x 6x 9 ( x 3) x 3x 9 (x 3) ( x 3) Step : Factor each denominator completely ( x 3) (x 3) (4x 6x Step : Write the product of all the different ( x 3) (x 3) (4x 6x Step 3: Raise each base to the highest exponent Revised /09 3

4 Example 9: Find the LCD of the expressions 4 3 ( x ) ( x) Solution: Denominators are ( x ) and its opposite are ( x) ( x ). Hence we move the 4 3 negative to the numerator ( x ) ( x ) We do not have to use any procedure to find the common denominator because both the fractions have the common denominator 7 Example 0: Find the LCD of the expressions 6( x 5) 4(5 x) Solution: (5 x ) is the negative of ( x 5) hence rewriting the expressions we have 7 6( x 5) 4( x 5) 6( x 5) 3 ( x 5) 4( x 5) ( x 5) Step : Factor each denominator completely 3 ( x 5) Step : Write the product of all the different 3 ( x 5) Step 3: Raise each base to the highest exponent Exercise: Find the Least Common Denominator (LCD) in each of the following exercise (Hint: follow the procedure suggested above). 3 x xy. 3x x x 6 3x x y y x 3 5. xy x x 3xy 6y a b 8b 8. 3n m nm a b a b x 36 Revised /09 4

5 5 9. c c 5 c 6c 9. y ( y 3) 5( y 3)( y y x 6 x x 3x 5. 4 x x 4 y 3 5 7y 7. 8y 6y 3y y 36 4y x 4 x 9. 3x x 6 x x a a a a 6. y 3y y 3y 3 8. x x x 4 3 x x 70 x 6x 9 6x 8 Solutions to the odd-numbered exercise and answers to the even- numbered exercise:. Denominators: x xy Step: x x xy x y Step: x y Step3: LCD x y. LCD : 4 3. Denominators x 6 3x Step: x ( x ) 6 3x 3 ( x ) Step : 3 ( x ) Step3: LCD 3 ( x ) Remark: opposite of (x-) is (-x) Hence we can rewrite the rational expression as 3 and find the LCD x 3x 6 4. LCD : ( y x ) or ( x y ) 5. Denominators xy x Step: xy x y x x Step : x y Step3: LCD x y 7. Denominators 3 8 x 3 xy6 y Step: 8x x 3xy 3 x y6y y Step : 3 x y Step3: LCD 3 x y 48 x y LCD :8ab LCD :3nm Revised /09 5

6 9. Denominators a b a b Step: a b ( a b) a a b b Step : a b ( a b) Step3: LCD a b ( a b). Denominators c c 5 c 6c 9 Step: c c 5 ( c 5)( c 3) c 6c 9 ( c 3) Step : ( c 5)( c 3) Step3: LCD ( c 5)( c 3) 3. Denominators 6x 6 x x Step: 6x 6 3 ( x x x x x x ( ( ( Step : 3 ( x ( x Step3: LCD 3 ( x ( x 5. Denominators 4 xx 4 Step x x y x x x : 4 ( ) 4 ( )( ) Step : ( x ) ( x ) Step3: LCD ( x ) ( x ) Remark: opposite of (x-) is (-x) Hence we can rewrite the fraction as 3x and find the LCD x 4 x 4 7. Denominators Step y y y y 8y 63y y 36 4y 3 :8 6 ( ) 3y y 36 3 ( y 3) ( y 4)4y ( y 3) Step : y ( y ) ( y 3) Step3: LCD y ( y ) ( y 3) 9. Denominators 3x x 6 x x 5 Step x x x x :3 6 (3 )( 3) x x 5 (x 5)( x 3) Step : (3x ) ( x 5) ( x 3) Step3: LCD (3x ) ( x 5) ( x 3) LCD : 3 5( x 4) or 45( x 4) LCD : 5( y 3) ( y LCD : ( a ( a 6. LCD : y( y 3)( y 3) LCD : ( x ( x ) LCD : 30( x 3) ( x 3)( x 3x Revised /09 6

7 Note: You can get additional instructions and practice for solving these problems by going to the following websites: This website has step-by-step instruction on how to find the least common multiple of integers and polynomials. Finding least common multiples is same as finding the least common denominators. This website has a you tube video on how to find the least common multiple which is same as find the least common denominator. ut0_addrat.htm#lcd This website provides video demonstration and step-bystep instruction on how to add two rational expressions. This website also has information on how to find the least common multiples Student friendly notes on adding and subtracting rational functions. This web site also has information on how to find the least common denominator of two or more fractions Revised /09 7