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INSTRUCTORS Carleen Eaton Grant Fraser Eric Smith
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Adding and Subtracting Rational Expressions with Unlike Denominators

  • The Least Common Multiple, or LCM, of a set of polynomials is the polynomial of smallest degree that is a multiple of each of the polynomials in the set.
  • To add or subtract fractions with unlike denominators, first find the LCM of the denominators. Then adjust the numerator and denominator of each fraction so that its denominator is the LCM. Finally, add or subtract the numerators and simplify the result.
  • Most of the time, the result will not simplify but always check to see if the numerator and denominator have any common factors.

Adding and Subtracting Rational Expressions with Unlike Denominators

Find the LCM of x2 − 11x + 30 and x2 − 25
  • Factor: x2 − 11x + 30 = (x − 5)(x − 6)
  • Factor: x2 − 25 = (x + 5)(x − 5)
LCM (x − 5)(x − 6)(x + 5)
Find the LCM of x2 + 6x − 16 and x2 − 4
  • Factor: x2 + 6x − 16 = (x − 2)(x + 8)
  • Factor: x2 − 4 = (x + 2)(x − 2)
LCM (x − 2)(x + 8)(x + 2)
Find the LCM of x2 − 3x − 70 and x2 − 49
  • Factor: x2 − 3x − 70 = (x − 10)(x + 7)
  • Factor: x2 − 49 = (x + 7)(x − 7)
LCM (x − 10)(x + 7)(x − 7)
Add
[2x/(4x − 4)] + [6/(5x − 5)]
  • [2x/(4(x − 1))] + [6/(5(x − 1))]
  • ( [5/5] )[2x/(4(x − 1))] + ( [4/4] )[6/(5(x − 1))]
  • [10x/(20(x − 1))] + [24/(20(x − 1))]
  • [(10x + 24)/(20(x − 1))]
  • [(2(5x + 12))/(2(10)(x − 1))]
[(5x + 12)/(10(x − 1))]
[3x/(5x + 15)] + [18/(7x + 21)]
  • [3x/(5(x + 3))] + [18/(7(x + 3))]
  • [21x/(35(x + 3))] + [90/(35(x + 3))]
[(21x + 90)/(35(x + 3))]
[4n/(2n − 14)] − [10/(9n − 63)]
  • [4n/(2(n − 7))] − [10/(9(n − 7))]
  • [36n/(18(n − 7))] − [20/(18(n − 7))]
  • [(36n − 20)/(18(n − 7))]
  • [(4(9n − 5))/(18(n − 7))]
[(2(9n − 5))/(9(n − 7))]
[4c/(6c − 36)] − [( − 8)/(4c − 24)]
  • [4c/(6(c − 6))] − [( − 8)/(4(c − 6))]
  • [8c/(12(c − 6))] − [( − 24)/(12(c − 6))]
  • [(8c + 24)/(12( c − 6 ))]
  • [(4( 2c + 6 ))/(4g3( c − 6 ))]
[(2c + 6)/(3( c − 6 ))]
[(x + 1)/(2x + 2)] − [(x + 3)/(2x2 − 8x − 10)]
  • [(x + 1)/(2x + 2)] − [(x + 3)/(( 2x + 2 )( x − 5 ))]
  • [(( x − 5 )g( x + 1 ))/(( x − 5 )g( 2x + 2 ))] − [(x − 3)/(( 2x + 2 )( x − 5 ))]
  • [(x2 − 4x − 5)/(( 2x + 2 )( x − 5 ))] − [(x + 3)/(( 2x + 2 )( x − 5 ))]
  • [(x2 − 4x − 5 − x − 3)/(( 2x + 2 )( x − 5 ))]
[(x2 − 5x − 8)/(( 2x + 2 )( x − 5 ))]
[(x + 1)/(3x − 1)] − [(3x + 12)/(3x2 − 11x − 4)]
  • [(x + 1)/(3x − 1)] − [(x + 12)/(( 3x − 1 )( x + 4 ))]
  • [(( x + 4 )g( x + 1 ))/(( x + 4 )g( 3x − 1 ))] − [(3x + 12)/(( 3x − 1 )( x + 4 ))]
  • [(x2 + 5x + 4)/(( 3x − 1 )( x + 4 ))] − [(3x + 12)/(( 3x − 1 )( x + 4 ))]
  • [(x2 + 5x + 4 − 3x − 12)/((3x − 1)( x + 4 ))]
  • [(x2 + 2x − 8)/(( 3x − 1 )( x + 4 ))]
[(( x − 2 )( x + 4 ))/(( 3x − 1 )( x + 4 ))] = [(x − 2)/(3x − 1)]
[(4y + 2)/(5y2 − 29y − 42)] + [(y − 8)/(( 5y + 6 ))]
  • [(4y + 2)/(( 5y + 6 )(y − 7)] + [(y − 8)/(5y + 6)]
  • [(4y + 2)/(( 5y + 6 )(y − 7))] + [((y − 8)g( y − 7 ))/((5y + 6)( y − 7 ))]
  • [(4y + 2)/(( 5y + 6 )(y − 7))] + [(y2 − 15y + 56)/((5y + 6)( y − 7 ))]
  • [(4y + 2 + y2 − 15y + 56)/((5y + 6)( y − 7 ))]
[(y2 − 11y + 58)/((5y + 6)( y − 7 ))]

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Adding and Subtracting Rational Expressions with Unlike Denominators

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Least Common Multiple of Polynomials 0:21
    • Example: Regular Fractions
    • Example: Rational Expressions
  • Equivalent Rational Expressions Using LCM 7:23
    • Example
  • Adding and Subtracting 14:24
    • Summary of Techniques
  • Example 1: Find the LCM 15:09
  • Example 2: Add Rational Expressions 17:53
  • Example 3: Subtract Rational Expressions 22:19
  • Example 4: Add Rational Expressions 30:44