Pubkeybreaker <pubkeybreaker@[EMAIL PROTECTED]
> wrote:
>On May 15, 1:17 am, Bill Dubuque <w...@[EMAIL PROTECTED]
> wrote:
>>Pubkeybreaker <pubkeybrea...@[EMAIL PROTECTED]
> wrote:
>>>On May 3, 8:57 pm, "michalc...@[EMAIL PROTECTED]
" <michalc...@[EMAIL PROTECTED]
> wrote:
>>>>
>>>> I tutor a lot of algebra and as you know there are three basic
>>>> types of problems that algebra students are given. [...]
>>>>
>>>> Example "Simplify (x^2-1)/(x+1)" this becomes "x-1".
>>>
>>> Which, of course, is just wrong. If x=-1, then the quotient is
>>> undefined, but the simplification equals -2. The correct answer
>>> must state this exception.
>>
>> No, (x^2-1)/(x+1) = x-1 is correct if the expressions denote
>> polynomials, which is quite frequently the case in "algebra".
>
> What have you been smoking? You seem to be reading things that
> are not there. I see no specification in the original problem
> that says "if the expressions are polynomials".
That is my point. The original problem statement doesn't specify
what the expressions denote. If they denote _formal_ polynomials,
then the inference is correct. Your criticism only holds true
if the expressions denote polynomial _functions_. Understanding
this form vs. function distinction is fundamental for algebra.
Indeed, once one learns how to suppress thinking of polynomials
only as functions, one gains access to the tremendous power
of universal techniques. For example, see [1] for application
to partial fraction decompositions, and see [2] for many further
applications (Cayley-Hamilton theorem, adj(A) = det(A)^(n-1),
Leibniz's product rule for polynomials, etc).
--Bill Dubuque
[1] http://google.com/groups?selm=y8zu0a6agwy.fsf%40nestle.csail.mit.edu
[2] http://google.com/groups?selm=y8zznru8v1d.fsf%40nestle.ai.mit.edu
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