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When is it necessary to find the least common denominator?

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When is it necessary to find the least common denominator (LCD) of two rational expressions? Describe, in your own words, the process for finding the LCD of two rational expressions. How is factoring related to this process? Give an example
use the following to explain.

p/p^2-3p+2+4/p-1

https://brainmass.com/math/basic-algebra/necessary-find-least-common-denominator-495158

SOLUTION This solution is FREE courtesy of BrainMass!

If you plan to do addition or subtraction of two rational expressions that have different denominators, then it is necessary to find the least common denominator (LCD) of two expressions in order to merge to a single expression.

For instance, if you want to simplify (x-5)/(x+4) - (x+2)/x, then you may need to find the LCD of (x+4) and x. This is easy, as the expressions (x+4) and x have NO common factors, the LCD is x(x+4).

To see if two expressions (say P(x), Q(x)) have common factors, it would be easy if we could factor out each of the two denominators. If there are common factors, we need to find the greatest common factor from both expressions P(x) and Q(x). Denote by F(x) the greatest common factor of P(x) and Q(x). Then LCD=P(x)Q(x)/F(x) or LCD=-P(x)Q(x)/F(x).

Now I can show you the process by using an example:

p/(p^2-3p+2)+4/(p-1)

First, we notice that the denominator of the expression p/(p^2-3p+2) is P=(p^2-3p+2), and the denominator of the expression 4/(p-1) is Q=(p-1).

Secondly, we factor out (p^2-3p+2): p^2-3p+2=(p-1)(p-2).

So, the greatest common factor of (p^2-3p+2) and (p-1) is F(p)=p-1.

In this case, F(p)=p-1. So, the LCD=P*Q/F=(p^2-3p+2)(p-1)/(p-1)=p^2-3p+2.

Now we need to change each expression to the one with the LCD as its denominator. So,

p/(p^2-3p+2) =p/[(p-1)(p-2)] ..................We did nothing.
4/(p-1) = [4(p-2)]/[(p-1)(p-2)] ...................We multiply both sides by (p-2)

p/(p^2-3p+2)+4/(p-1)

=p/[(p-1)(p-2)] + [4(p-2)]/[(p-1)(p-2)]

=[p+4(p-2)]/[(p-1)(p-2)]

=(5p-8)/[(p-1)(p-2)]

This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!