Date: Jan 16, 2013 3:49 AM
Author: William Elliot
Subject: Re: simplifying rational expressions

On Tue, 15 Jan 2013, stony wrote:
>
> Need a little help with this. We are simplifying the following, but
> the solution is pretty lengthy and messy because of the enormous
> number of factors. I was thinking that may be I am missing seeing a
> pattern (some series or something). Is grunt work the only way to
> solve this or is there a pattern that can simplify the whole process?
>

Write very neatly and work methodically cheching and doubling
cheching each step as you go and again afterwards.

> ((b-c)/((a-b)(a-c))) + ((c-a)/((b-c)(b-a))) + ((a-b)/((c-a)(c-b))) +
> (2/(b-a)) - (2/(c-a))
>
> of course, I took all the factors in the denominator and then started
> multiplying the numerator with the remaining factors to end up with a
> mess.


-(b-c)/((a-b)(c - a)) - (c-a)/((b-c)(a - b))) - (a-b)/((c-a)(b - c)))
- 2/(a - b) - 2/(c-a)

-[(b - c)^2 + (c - a)^2 + (a - b)^2 + 2(b - c)(c - a) + 2(a - b)(c - a)]

-[b^2 - 2bc + c^2 + c^2 - 2ac + a^2 + a^2 - 2ab + b^2
+ 2(bc - c^2 + ab + ac) + 2(ac - bc - a^2 + ab)]

-[b^2 - 2bc + 2ab + 2ac]