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Math Help - why long division of a polynomial is the same as the value.

  1. #1
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    why long division of a polynomial is the same as the value.

    Hi,

    An example:
    To calculate f(5) for f(x)=x^3-3x^2+7x-4 I can either subsitute 5 for x and get 81, or I can divide the polynomial by (x-5) and evaluate the remainder to get the same value.

    This is really quite astounding. My reference(s) don't explain why this is true. I have given it some thought, but cannot figure out why this symetry (if it is a symetry) is true.

    Any ideas?

    Regards
    Craig.
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  2. #2
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  3. #3
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    If \frac{P(x)}{x- a} is equal to Q(x) with remainder r, then \frac{P(x)}{x-a}= Q(x)+ \frac{r}{x-a} or, multiplying both sides by x- a, P(x)= Q(x)(x- a)+ r.

    Taking x= a, P(a)= Q(a)(a- a)+ r= Q(a)(0)+ r= r.
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