Comments

Dividing each side by (x-5) would ultimately "prove" that 1=0, not that 0=0.

Correct, of course. My mistake -- but just a typo. However, that should give a hint to the hidden flaw in this method.

There's a big problem here. Let's say you wanted to solve (x-1)(x-2)=0. by your method, you could divid both sides by (x-1), and get x=2 as the answer. but what if x=1? then, you can't divide both sides by (x-1). contrary to popular belief, you can't even divide ZERO by zero. so your answer is x=1 or x=2. in your very first example, what if (353i)/(x^3+ 23) equals zero? then you'd be dividing by zero. again, that's not allowed. another solution to this problem would be to say that 353i/(x^3+23) EQUALS zero. when you use this annihilator method, you're not getting rid of hassle, you're getting rid of a solution. what you CAN do, however, is when you've got a bunch of things multiplying together to equal zero, is just say that the first thing, or the second thing, or the third thing, MUST equal zero for the entire product to equal zero. let me give you a hint: that's already been done before. how did you learn to do the problem x^2 - 5x + 6 = 0 ? you factor it: (x-2)(x-3)=0. then, without thinking, you jump a step and conclude (correctly) that x=2 or x=3. what you're missing there is the intermediate step:

(x-2)=0 OR (x-3)=0

biotch.

You've nailed it. The trouble is that dividing by zero is eternally a no-no, so if you divide out a term whose value you don't know, it's possible that term is zero, you're therefore dividing zero by zero, and breaking the LAWS OF THE UNIVERSE.

Practically speaking, since nobody really cares about the laws of the universe, the only real problem with this is that it ends up hosing solutions. "Equalling zero" is another word for "root" which is another way to say "solution," so if you take out what happens to be a zero, you've removed what could be a necessary solution.

The reason I don't consider this to be a dealbreaker is because you don't always NEED every single solution -- sometimes you only need one -- and moreover, you aren't necessarily even dividing out possible solution sets. For instance, in my example, maybe the whole "kzy" term is useless to you, and has no relevance to the needed answer. Cool; you can get rid of it without fear, and focus on the x's.