Difference between revisions of "2006 SMT/Algebra Problems/Problem 4"
(Created page with "==Problem== Simplify: <math> \frac{a^3}{(a-b)(a-c)}+\frac{b^3}{(b-a)(b-c)}+\frac{c^3}{(c-a)(c-b)} </math> ==Solution== Firstly, we notice that the factors in the denominators ar...") |
(No difference)
|
Latest revision as of 08:55, 28 May 2012
Problem
Simplify:
Solution
Firstly, we notice that the factors in the denominators are almost the same. To make them the same, we merely have to change a couple of signs, to get
Now it becomes easier to combine the fractions. We have
Now we try expanding the numerator.
Now we can try to factor the numerator. Notice that letting or in the numerator causes it to become , so is a factor of the numerator. Therefore, we only need one more degree of and in the numerator. We first try as the remaining factor in the numerator, and see that it works. Therefore,