Intermediate Algebra 07: Finding the Lowest Common Denominator

HFCC Math Lab Intermediate Algebra - 7FINDING THE LOWEST COMMON DENOMINATOR (LCD)Adding or subtracting two rational expressions require the rational expressions to have thesame denominator.Example 1: Adding two rational expressions with the common denominator 2x+15 32x1 2x1= 5 32x18=2x1When the denominators of two rational expressions are not the same we write equivalentrational expression for both or one of the rational expressions so that they both have samedenominator.To find the equivalent rational expressions we multiply the numerator and denominator of arational function with same non-zero factor.Example 2: A list of equivalent rational expressions to 2 x 15x12x1 1010(2x1)=>5x1 1010(5x1)2x15x1xx=>x(2x1)x(5x1)assuming x 02x1 (2x3)5x1 (2x3)=>(2x1)(2 x 3)(5x 1)(2 x 3)assuming 2x3 022x 1 10 x (3x5)25x 1 10 x (3x5)210 x (3x 5)(2x1)210 (3 5)(5 1)=>x x x assuming 210 x (3x5) 0Remark: Try to reason out why every rational expression in the list above is equivalent to 2 x 15x1We write equivalent rational expressions so that the two rational expressions have commondenominator 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 yourknowledge on equivalent rational expressions to do so).Revised 11/09 1

Example 3: Some of the multiples of the polynomial26 x(5x 1) are16 x(2x 1) 6 x(2x1)2 236 x(2x 1) 18 x(2x1)2 2x 6 x(2x 1) 6 x (2x1)5 2 6 25(2x 1) 6 x(2x 1) 30 x(2x1)2 37 x (2x 1) 6 x(2x 1) 42 x (2x1)2 3 2 3 5Remark: Try to reason out why every expression in the above list is a multiple of26 x(5x1)The best choice for the denominator is the Least Common Multiple (LCM) of thedenominators of two expressions. We also refer to this value as the Least CommonDenominator (LCD)Remark: Try to reason why the above statement makes sense. (Hint: Least means smallest multipleof a polynomial. A smallest multiple of a polynomial has the least degree and yet can be written as aproduct of both the polynomials under consideration.)To find the least common denominator (LCD) of two or more fractional expression youcan follow this procedure:1. Factor each denominator completely, showing repeated factors as powers; i.e., asbases with exponents.2. Write the product of all the different bases without exponents.3. Raise each base to the highest exponent to which it is raised in any singledenominatorRemark: A base and its opposite (negative) can be considered the same when find theLCD; any (-1) factor can be attached to the numerator.Example 4: Simplifying rational expression with a -1 factor in the denominator1 1( x 2) x 2Example 5: Find the Least Common Denominator (LCD) of the expressionsSolution: Using the above procedure to find the LCDx 1 x 4 x, ,20 6 102022 56 23 10 25 23 5Step 1: Factor each denominator completely,showing repeated factors as powersStep 2: Write the product of all the differentbases without exponents.22 3 5 60Step 3: Raise each base to the highest exponentto which it is raised in any single denominatorRevised 11/09 2

1 5xy 7Example 6: Find the Least Common Denominator (LCD) of the expressions , ,2 2 318x y 10xy z 15yzSolution: Using the above procedure to find the LCD218x y210xy z215yz2 2 223 x y25x y35 yz3 zStep 1: Factor each denominator completely,showing repeated factors as powers23 5 x yzStep 2: Write the product of all the differentbases without exponents.2 3 5 902 2 3 2 3 x y z x y z Step 3: Raise each base to the highest exponentto which it is raised in any single denominator4 3 7Example 7: Find the Least Common Denominator (LCD) of the expressions , ,2 2a 1 2a 2 ( a 1)Solution: Using the above procedure to find the LCD2a a a1 ( 1) ( 1)2a2 2 ( a 1)2 2( a 1) ( a 1)Step 1: Factor each denominator completely,showing repeated factors as powers2 ( a 1) ( a 1)Step 2: Write the product of all the differentbases without exponents.22 ( a 1) ( a 1) Step 3: Raise each base to the highest exponentto which it is raised in any single denominator4 3 7Example 8: Find the LCD of the expressions , ,3 2 28x 27 x 6x 9 2x 3x9Solution: Using the above procedure to find the LCD3 28x 27 (2x 3) (4 x 6x9)x 6x 9 ( x 3)2 222x 3x 9 (2x 3) ( x 3)Step 1: Factor each denominator completely,showing repeated factors as powers2( x 3) (2x 3) (4x 6x9) Step 2: Write the product of all the differentbases without exponents.2 2( x 3) (2x 3) (4x 6x9) Step 3: Raise each base to the highest exponentto which it is raised in any single denominatorRevised 11/09 3

Example 9: Find the LCD of the expressions4 3,( x 2) (2 x)Solution: Denominators are ( x 2) and its opposite are (2 x) 1( x 2) . Hence we move the4 3negative to the numerator ,( x 2) ( x 2)We do not have to use any procedure to find the common denominator because both the fractionshave the common denominator7 1Example 10: Find the LCD of the expressions ,26( x 5) 4(5 x)Solution: (5 x ) is the negative of ( x 5) hence rewriting the expressions we have7 1,26( x 5) 4( x 5)6( x 5) 23 ( x 5)2 224( x 5) 2 ( x 5)Step 1: Factor each denominator completely,showing repeated factors as powers2 3 ( x 5)Step 2: Write the product of all the differentbases without exponents.2 22 3 ( x 5)Step 3: Raise each base to the highest exponentto which it is raised in any single denominatorExercise: Find the Least Common Denominator (LCD) in each of the following exercise(Hint: follow the procedure suggested above)1.2 3,x xy2.3x2 x 4,8 123.1 3,x 2 6 3x4.3 5,4x 2y y 2x1 35. ,22xy x5 4 17. , ,2 38x 3xy 16y3 56. ,2 24a b 8b1 28. ,2 23n m nm9.9 4 3, ,a b a b10.3 8,5x20 9x36Revised 11/09 4

5 911. ,2 2c 2c 15 c 6c912.2y,2( y 3) 5( y 3)( y 1)y21 2 313. , ,26x 6 x 1 2x23x115. ,24 2x x 4y 3 5 7y17. , ,2 28y 16y 3y 21y 36 4y12x 4 x19. ,2 23x 11x 6 2x x 151 2 314. , ,2 2a 1 a 2a 1 2a21 116. ,2 2y 3y y 3y3 118. ,2 2x x 2 x 43 x 120. , ,3 210x 270 x 6x 9 6x18Solutions to the odd-numbered exercise and answers to the even- numbered exercise:1. Denominators: x,xyStep1: x x,xy xyStep2:xyStep3:LCD xy2. LCD : 243. Denominators x 2,6 3xStep1: x 2 ( x 2), 6 3x 3 ( x 2)Step2 : 3 ( x 2)Step3: LCD 3 ( x 2)Remark: opposite of (x-2) is (2-x)Hence we can rewrite the rational expression as1 3, and find the LCDx 2 3x64. LCD : 2( y 2 x ) or 2(2 x y )5. Denominators 2 xy,xStep1: 2xy 2 x y,x xStep2 : 2x y2Step3: LCD 2x y7. Denominators22 22 38 x ,3 xy,16yStep1:8x 2 x ,3xy 3 x y,16y 2 yStep2 : 23x y2 3 2 3 4 3Step3: LCD 2 3 x y 48x y4 2 3 2 36.8.LCD :8a b2 22 2LCD :3n mRevised 11/09 5

9. Denominators a b, a,bStep1: a b ( a b), a a,b bStep2 : ab ( a b)Step3: LCD ab ( a b)11. Denominators2 2c 2c 15, c 6c9Step1: c 2c 15 ( c 5)( c 3), c 6c 9 ( c 3)2 2 2Step2 : ( c 5)( c 3)Step3: LCD ( c 5)( c 3)213. Denominators 6x 6, x 1,2 x 2Step1: 6x 6 23 ( x 1),x x x x x21 ( 1),2 2 2 ( 1) ( 1)Step2 : 23 ( x 1) ( x 1)Step3: LCD 23 ( x 1) ( x 1)15. Denominators24 2 xx , 4Step x x y x x x21: 4 2 2 ( 2) , 4 ( 2)( 2)Step2 : 2 ( x 2) ( x 2)Step3: LCD 2 ( x 2) ( x 2)Remark: opposite of (x-2) is (2-x)Hence we can rewrite the fraction as3x1, and find the LCD22x4 x 417. DenominatorsStep y y y y22 28y 16,3y 21y 36,4 y 122 31:8 16 2 ( 2),2 23y 21y 36 3 ( y 3) ( y 4),4y 12 2 ( y 3)Step2 : 2 y( y 2) ( y 3)Step3: LCD 2 y( y 2) ( y 3)19. Denominators2 23x 11x 6,2x x 15Step x x x x21:3 11 6 (3 2)( 3),22x x 15 (2x 5)( x 3)Step2 : (3x 2) (2 x 5) ( x 3)Step3: LCD (3x 2) (2 x 5) ( x 3)10.12.14.2LCD : 3 5( x 4) or 45( x 4)2LCD : 5( y 3) ( y 1)2LCD : 2( a 1)( a 1)16. LCD : y( y 3)( y 3)18.20.2LCD : ( x 1)( x 2)2 2LCD : 30( x 3) ( x 3)( x 3x9)Revised 11/09 6

Note: You can get additional instructions and practice for solving theseproblems by going to the following websites:http://www.purplemath.com/modules/lcm_gcf.htm This website has step-by-stepinstruction on how to find the least common multiple of integers andpolynomials. Finding least common multiples is same as finding the leastcommon denominators.http://www.youtube.com/watch?v=drZopvFRa7s This website has a you tubevideo on how to find the least common multiple which is same as find the leastcommon denominator.http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut10_addrat.htm#lcd This website provides video demonstration and step-bystepinstruction on how to add two rational expressions. This website also hasinformation on how to find the least common multipleshttp://www.regentsprep.org/Regents/math/algtrig/ATO2/addfrac.htm Studentfriendly notes on adding and subtracting rational functions. This web site alsohas information on how to find the least common denominator of two or morefractionsRevised 11/09 7