SOLUTION: what is the least common denominator of one over p minus four and two over four minus p?

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Question 819710: what is the least common denominator of one over p minus four and two over four minus p?
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
I'm assuming that the expressions are:
1%2F%28p-4%29 and 2%2F%284-p%29
Please put parentheses around numerators and denominators if they are not just a positive integer or variable. So if I am right about your expression then you should have posted:
"... one over (p minus four) and two over (four minus p)"

The answer to this question is tricky. I can show you three different answers, all of which (as far as I know) could be called the lowest common denominator.

Lowest common denominator (LCD):
  • The greatest common factor between the two denominators is 1. When this is true the LCD is the product of the two denominators: (p-4)(4-p). Multiplying we get the LCD:
    -p%5E2%2B8p-16
  • If we recognize that the two denominators are exact opposites of each other then we can come up with simpler LCD's by multiplying the numerator and denominator of one of the fractions by -1/-1:
    • %281%2F%28p-4%29%29%28%28-1%29%2F%28-1%29%29
      %28-1%29%2F%28-p%2B4%29
      Reordering the denominator:
      %28-1%29%2F%284%2B%28-p%29%29
      Rewriting as a subtraction:
      %28-1%29%2F%284-p%29
      This makes (4-p) the LCD.
    • %282%2F%284-p%29%29%28%28-1%29%2F%28-1%29%29
      %28-2%29%2F%28-4%2Bp%29
      Reordering the denominator:
      %28-2%29%2F%28p%2B%28-4%29%29
      Rewriting as a subtraction:
      %28-2%29%2F%28p-4%29%29
      This makes (p-4) the LCD.
Personally I prefer the last one: (p-4) because it is simpler and because the variable has a positive coefficient.