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Chapter 8: Rational Expressions

8.3 Least Common Denominators

Finding the least common denominator, or LCD, is very important to working with rational expressions. The process used depends on finding what is common to each rational expression and identifying what is not common. These common and not common factors are then combined to form the LCD.

Example 8.3.1

Find the LCD of the numbers 12, 8, and 6.

First, break these three numbers into primes:

12=223 or 2238=222 or 236=23

Then write out the primes for the first number, 12, and set the LCD to 223.

Notice the factorization of 8 includes 23, yet the LCD currently only has 22, so you add one 2.

Now the LCD = 233.

Checking 6=23, there already is a 23 in the LCD, so we need not add any more primes.

The LCD = 233 or 24.

This process can be duplicated with variables.

Example 8.3.2

Find the LCD of 4x2y5 and 6x4y3z6.

First, break both terms into primes:

4x2y5=22x2y56x4y3z6=23x4y3z6

Then write out the primes for the first term, 4x2y5, and set the LCD to 22x2y5.

The LCD for 6x4y3z6=23x4y3z6 has an extra 3, x2, and z6, which we add to the LCD that we are constructing.

This yields LCD = 223x2+2y5z6, or LCD = 12x4y5z6.

This process can also be duplicated with polynomials.

Example 8.3.3

Find the LCD of x2+2x3 and x2x12.

First, we factor both of these polynomials, much like finding the primes of the above terms:

x2+2x3=(x1)(x+3)x2x12=(x4)(x+3)

The LCD is constructed as we did before, except this time, we write out the factored terms from the first polynomial, so the LCD = (x1)(x+3).

Notice that x2x12=(x4)(x+3), where the (x+3) is already in the LCD, which means that we only need to add (x4).

The LCD = (x1)(x+3)(x4).

Questions

For Questions 1 to 10, find each Least Common Denominator.

  1. 2a3,6a4b2,4a3b5
  2. 5x2y,25x3y5z
  3. x23x,x3,x
  4. 4x8,x2,4
  5. x+2,x4
  6. x,x7,x+1
  7. x225,x+5
  8. x29,x26x+9
  9. x2+3x+2,x2+5x+6
  10. x27x+10,x22x15,x2+x6

For Questions 11 to 20, find the LCD of each fraction and place each expression over the same common denominator.

  1. 3a5b2,210a3b
  2. 3xx4,2x+2
  3. x+2x3,x3x+2
  4. 5x26x,2x,3x6
  5. xx216,3xx28x+16
  6. 5x+1x23x10,4x5
  7. x+1x236,2x+3x2+12x+36
  8. 3x+1x2x12,2xx2+4x+3
  9. 4xx2x6,x+2x3
  10. 3xx26x+8,x2x2+x20,5x2+3x10

Answer Key 8.3

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