Thread: why long division of a polynomial is the same as the value.

1. 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.

2. 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, $P(a)= Q(a)(a- a)+ r= Q(a)(0)+ r= r$.