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Re: Factoring
Posted:
Jan 9, 2012 1:15 AM
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"John Smith" <invalid@invalid.invalid> wrote in message
news:jedk9k$dna$1@speranza.aioe.org...
> "Peter Webb" <r.peter.webbbbb@gmail.com> wrote in message
> news:jedieh$87h$1@news.albasani.net...
>>
>> "John Smith" <invalid@invalid.invalid> wrote in message
>> news:jedg18$6c8$1@speranza.aioe.org...
>>>A student asked me how to factor x^3-3x^2-4x+12
>>> So I took a guess (correctly it seems) that since 12 factors as 2,2,3
>>> one of the factors was likely to be x-2. Dividing by x-2 gives x^2-x-6
>>> which is easy to factor into (x-3)(x+2) so the answer is
>>> (x-2)(x-3)(x+2).
>>>
>>> Is there a way to do that whithout guesswork?
>>>
>>> Suppose I had a more complicated example such as (2x+1)(3x-2)(4x+5) I
>>> haven't multiplied it out but how would I find the factors if I didn't
>>> know them in advance?
>>>
>>
>> http://en.wikipedia.org/wiki/Cubic_function
>>
>> Which may explain why guessing the answer (usually by finding factors of
>> the constant term) is so popular. The alternative is awful. For quartic
>> equations (ax^4 + bx^3 + .. + e) it is even worse, for quintics it is
>> non-existent.
>
> Yes ok I guess I already had that at the back of my mind but was hoping to
> find some kind of procedure or algorithm for finding the answers to
> factoring problems which are likely to appear in current school text
> books.
>
You already know the algorithm to solve schoolbook exercises. Look at the
factors of the constant term. This will always lead to a solution if the
solutions are all integers.
Cubics in school textbooks almost always have integer solutions, and this
approach (or a minor variation) will almost always work. Cubics in real life
almost never have integer solutions, and this approach seldom works. As
cubics are mostly taught as examples of polynomials, and that is what is
really being studied, it doesn't matter if the cubics used as examples are
easier to solve than they typically are in real life.
> There are web sites which can factor any natural number but it's a mystery
> to me how they do it.
>
> I found out many decades ago that the real number solutions for some roots
> of some cubics require a trip through complex numbers. As far as quintics
> are concerned I'll probably never understand Galois theory.
>
Cubic equations are what mostly convinced mathematicians that complex
numbers existed in some sense and could be studied like "Real" numbers. That
is history, not mathematics; they were starting to pop up in a few places,
it was an idea whose time had come.
> old guy
>
>>
>>> Thanks for any enlightenment.
>>>
>>> old guy
>>>
>>
>
>
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