Posted by T.Gracken on October 27, 2002 at 07:34:03:
In Reply to: nomenclature question posted by Denis Borris on October 26, 2002 at 17:47:51:
: why in hell did they call it "the remainder theorem"?!
: really misleading for someone who comes across it for 1st time.
: ax^3 + bx^2 + cx + d: instead of calling it the remainder theorem,
: a name that represents: "what is "d" short or over by" would be much
: clearer; agree??
no. the problem with coming across it for the first time (at least here) is that the background has not been presented.
The idea is if you have a polynomial divided by a first degree binomial, then can we determine if the polynomial has a factor equal to the binomial divisor.
i.e. does the polynomial factor where one of the factors is the divsor.
From the division algorithm f(x)/d(x) = q(x) + r(x)/d(x) [f,d,q,r polynomials], it can be concluded that the polynomial can be factored if and only if the remainder, r(x), equal zero. We are considering more than just the "d" you wrote above. It may (or may not) be affected by preceding terms when dividing.
The remainder theorem then states that if f is a polynomial and d(x)=x-k [first degree binomial] then the remainder r(x)=f(k).
That is, with some basic evaluating (constants only), we can determine what the 'remainder' of the division problem will be.
Why? because if it is zero, then we can factor the polynomial.
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