Factoring PolynomialsDate: Wed, 16 Nov 1994 From: Anonymous Subject: Factoring I am in need of an easy method to understanding factoring polynomials, especially with exponents. If you can, touch a little on quadratic equations also. Thanks. Mark Muscovitch markm@epix.net Date: Wed, 16 Nov 1994 16:46:15 -0500 (EST) From: Dr. Ken Subject: Re: Factoring Hello there! Ah, factoring polynomials. In my recollection, I spent pretty much my entire 8th grade getting good at factoring polynomials. My friend Ethan (another Math Doctor here) spent three or four years on it, so he's really good! My first instinct is to tell you that the only way to really get good at factoring polynomials is to do it a whole bunch of times. I know that sounds like a dumb and cheap way out, but I think it's true. Anyway, I'll tell you something else. Let's say we want to factor the polynomial (3x + 2)(4x - 3) = 12x^2 - x - 6. I'm sure you've been taught a couple of ways to do it in school, but there's one foolproof way that always works (if your polynomial factors at all). You can use the Quadratic Formula. The problem is that you sometimes get messy numbers when you use the Quadratic Formula, but sometimes you don't. Let's see how it applies here: ___________ we know that -b +- \/b^2 - 4ac x = ___________________________ 2a So we get, in our problem, _____________ 1 +- \/1 - 4(12)(-6) 1 +- 17 3 -2 x = ___________________________ = ___________ = ___ or ___ 24 24 4 3 / 3 \/ 2 \ |x - ___ ||x + ___ | So we think we can write our polynomial as \ 4 /\ 3 / . But look out. You'll notice that if you actually multiply that out, you don't get the right answer. What you DO get, though, is the answer divided by 12. See that? So what we'll do is take what we got and multiply it by 12, and we'll get (4x - 3)(3x + 2) when the dust settles. Now THAT's the real answer. Was it worth it? Could we just have made a table of all the divisors of 12, and all the divisors of 2, and just tried with our bare hands to get the answer? Sure. As a matter of fact, it would have been quicker. But try to factor this one: x^2 + x + 1. You won't be able to do it. It doesn't factor. The reason is that if you look at its graph, it'll never cross the x-axis. So you can't look at its roots the way we did above. What about this one: 3x^ - 2x + 17 ? Is it messy? Is it nice? I hope this helps. More to the point, I hope I didn't confuse you even more. Thanks for the question! -Ken "Dr." Math |
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