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Thread: 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 $\displaystyle f(5)$ for $\displaystyle f(x)=x^3-3x^2+7x-4$ I can either subsitute $\displaystyle 5$ for $\displaystyle x$ and get 81, or I can divide the polynomial by $\displaystyle (x-5)$ 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 $\displaystyle \frac{P(x)}{x- a}$ is equal to Q(x) with remainder r, then $\displaystyle \frac{P(x)}{x-a}= Q(x)+ \frac{r}{x-a}$ or, multiplying both sides by x- a, $\displaystyle P(x)= Q(x)(x- a)+ r$.

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