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MathGroup Archive 2002

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Re: Re: Re: Factoring question

  • To: mathgroup at smc.vnet.net
  • Subject: [mg35517] Re: [mg35485] Re: Re: Factoring question
  • From: Andrzej Kozlowski <andrzej at yhc.att.ne.jp>
  • Date: Wed, 17 Jul 2002 02:09:10 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

Mathematica does not try to use any term groupings or re-arrangings that 
people usually try, because for computers much more direct methods are 
much faster and also guarantee getting the answer. In this case the 
expression is a quartic, so Mathematica can solve it in radicals using 
the formula of Cardano. The easiest way for a human who can't remember 
such complicated things is  to notice that p  and q are obviously roots 
of y^4 - (p + q)*y^3 + (p^2*q + p*q^2)*y - p^2*q^2==0. Thus (x-p)(x-q) 
must be a factor a factor. Once you know that the of the problem is 
trivial. Using Mathematica you can finish it  in lots of ways, e.g.

In[1]:=
Cancel[((-p^2)*q^2 + (p^2*q + p*q^2)*y - (p + q)*y^3 + y^4)/
    ((-p + y)*(-q + y))]

Out[1]=
(-p)*q + y^2

or

In[2]:=
PolynomialQuotient[y^4 - (p + q)*y^3 + (p^2*q + p*q^2)*y -
    p^2*q^2, (y - p)*(y - q), y]

Out[21]=
(-p)*q + y^2


or

In[3]:=
SolveAlways[y^4 - (p + q)*y^3 + (p^2*q + p*q^2)*y -
     p^2*q^2 == (a*y^2 + b*y + c)*((y - p)*(y - q)), y]

Out[3]=
{{c -> (-p)*q, a -> 1, b -> 0}, {c -> 0, a -> 1, b -> 0,
    p -> 0}, {c -> 0, a -> 1, b -> 0, q -> 0}}

The last method is equivalent to what is sometimes called "comparing 
coefficients".


On Tuesday, July 16, 2002, at 04:49  AM, Steven Hodgen wrote:

> Actually, I never posted the actual problem, but here it is:
>
> y^4 - (p + q)*y^3 + (p^2*q + p*q^2)*y - p^2*q^2
>
> This factors to:
>
> (y^2 - p*q)*(y - p)*(y - q)
>
> Which I cannot get.  I've tried over an over again, and I'm the best
> factorer in my class, my instructor can't seem to get this either, 
> hence my
> hope that Mathematica could do it, and it certainly can no problem, but 
> it
> just doesn't show how.  So, any ideas?
>
> --Steven
>
> "Ken Levasseur" <Kenneth_Levasseur at uml.edu> wrote in message
> news:agrjfj$g9n$1 at smc.vnet.net...
>> Steve:
>>
>> I assume that the problem was  to solve x^7 + x^5 + x^4 + -x^3 + x + 
>> 1=0.
> If
>> so, one of the basic factoring theorems is that if a polynomial over 
>> the
>> integers  like this one has a rational root r/s, then r must divide the
>> constant term and s must divide the leading coefficient.   So in this
> problem,
>> +/-1 are the only possible rational roots and so the (x+1) factor 
>> would be
>> found this way.   I'm sure that Mathematica checks this almost
> immediately.
>> As for the remaining 6th degree factor, I'm not certain how Mathematica
>> proceeds, but if you plot it, it clearly has no linear factors.
>>
>> Ken Levasseur
>>
>>
>> Steven Hodgen wrote:
>>
>>> "DrBob" <majort at cox-internet.com> wrote in message
>>> news:agbfhl$je9$1 at smc.vnet.net...
>>>> Factor[x^7 + x^5 + x^4 + -x^3 + x + 1] // Trace
>>>
>>> This doesn't do it.  It only traces the initial evaluation, and then
> simply
>>> displays the factored result with no intermediate factoring steps.
>>>
>>> Thanks for the suggestion though.
>>>
>>>>
>>>> Bobby
>>>>
>>>> -----Original Message-----
>>>> From: Steven Hodgen [mailto:shodgen at mindspring.com]
To: mathgroup at smc.vnet.net
>>>> Subject: [mg35517] [mg35485]   Factoring question
>>>>
>>>> Hello,
>>>>
>>>> I just purchased Mathematica 4.1.  I'm taking precalculus and wanted
> to
>>> try
>>>> a tough factoring problem, since the teacher couldn't do it either.
>>>> Mathematica get's the correct answer, but I'm interrested in seeing
> how it
>>>> got there.  Is there a way to turn on some sort of trace feature 
>>>> where
> it
>>>> shows each step it used to get the the final result?
>>>>
>>>> Thanks!
>>>>
>>>> --Steven
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>
>>
>
>
>
>
>
Andrzej Kozlowski
Toyama International University
JAPAN
http://platon.c.u-tokyo.ac.jp/andrzej/



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