Thread: The Golden Section. View Single Post 2004-04-27, 12:35   #13
R.D. Silverman

Nov 2003

22·5·373 Posts Quote:
 Originally Posted by jinydu The problem is with the third line (x = 2). I wanted to simplify cbr(10 + sqrt(108)) + cbr(10 - sqrt(108)) without knowing a priori that the answer is 2. That way, it would be possible to generalize the technique to cases where I don't know what x equals to beforehand.
This just begs the question. When you say "simplify" what does this mean?
If you mean "can it be simplified to an integer and if so how?", the answer
is obvious. Even a very rough mental calculation shows that *if* the answer
is an integer it must be near cube root of 20, thus it must equal 2 or 3.
If you further look at the original equation you realize that if the root is an
integer it must be a divisor of 20 (the product of the roots is 20). Thus
we are led to x = 2. This is not guessing. Since the polynomial is monic,
we know that if there is a rational root, it will be an integer.

What else are you looking for? You have not defined what "simplify" means.
You must state what form you expect the final answer to take before you
can "simplify".  