SOLUTION: Find the LCD of the rational expressions in the list. {{{(h)/(h^2-k^2)}}},{{{(9)/(h^2-2hk+k^2)}}} thank you for the help.

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Find the LCD of the rational expressions in the list. {{{(h)/(h^2-k^2)}}},{{{(9)/(h^2-2hk+k^2)}}} thank you for the help.      Log On


   

Question 425608: Find the LCD of the rational expressions in the list.
%28h%29%2F%28h%5E2-k%5E2%29,%289%29%2F%28h%5E2-2hk%2Bk%5E2%29


thank you for the help.
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
To find the LCD we need to know what the factors of the denominators are. So we start by factoring the denominators. (From this point on I am going to ignore the numerators since they have no impact of what the LCD is or on how we find the LCD.)

The first denominator is a difference of squares so we can use the a%5E2-b%5E2+=+%28a%2Bb%29%28a-b%29 pattern to factor it:
h%5E2-k%5E2+=+%28h%2Bk%29%28h-k%29

The second denominator is a perfect trinomial square so we can use the a%5E2-2ab%2Bb%5E2+=+%28a-b%29%28a-b%29 pattern tp factor it:
h%5E2-2hk%2Bk%5E2+=+%28h-k%29%28h-k%29

So our two denoinators, in factored form are:
(h+k)(h-k) and (h-k)(h-k)

The LCD will be the smallest product that includes all the factors of both denominators. In this case the LCD will be:
(h+k)(h-k)(h-k)
Looking at the LCD you can see the first denominator in the first two factors and the second denominator in the last two factors.

(h+k)(h-k)(h-k)
is the LCD and this may be an acceptable form. If not, then multiply this out. Multiplying the first two factors is easy because they cam from the first denominator:
%28h%5E2-k%5E2%29%28h-k%29
Now we just use FOIL:
h%5E3-h%5E2k-hk%5E2-k%5E3

So the LCD is either
(h+k)(h-k)(h-k)
or
h%5E3-h%5E2k-hk%5E2%2Bk%5E3
depending on the desired form.