> 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? > > My daughter was trying to solve this, but ended up with the mess and > then I got the same mess, but I thought there may be an easy way to > simplify this that I may be missing. > > ((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.
I suspect you are missing the fact that (a - b) and (b - a) in the denominators are essentially the same factor. The common denominator of the terms is (a - b)(a - c)(b - c). I think you can leave the denominator in that form.
I did the numerator two ways: in terms of these same factors, and multiplying out and combining terms. They seemed about equally simple to me, though I may have made mistakes. I usually do. Be careful with the signs :)