Why does dividing a polynomial by $x-a$ give the same quotient as evaluating it at $x=a$ using synthetic division? I know synthetic division is a table representation of the calculations that occur when evaluating a polynomial in Horner's form at some $x$ value ($x=a$). 
For example, if there's some polynomial in standard form say $x^3 + 4x^2 -5x + 5$ it can be transformed into Horner's form through successive groupings and factoring out $x$. 
$x^3 + 4x^2 -5x + 5$ 
$=(x^2+4x-5)x+5$
$=((x+4)x-5)x+5$ 
If this polynomial is evaluated at $x=3$ the result is 
$p(3) =(((3)+4)(3)-5)(3)+5$ 
$=((7)(3)-5)(3)+5$ 
$=(21-5)(3)+5$ 
$=(16)(3)+5$ 
$=48+5$ 
$=53$
Which is also what we get using synthetic division

Interestingly, synthetic division also gives the same quotient as dividing $x^3 + 4x^2 -5x + 5$ by $x-3$ 

Which of course is $x^2 + 7x + 16 +\frac{53}{x-3}$
My question are: 
1) Why would dividing a polynomial by $x-a$ (using long division) give the same quotient as evaluating it at $x = a$ when using synthetic division? 
2) Furthermore why are only the coefficients and $a$ term used in synthetic division? What happened to the variables and their respective powers? They seem to "disappear" from the calculation when using synthetic division yet the same quotient is derived just as using long division. 
 A: It is easy to understand why the remainder obtained upon 'dividing' by $(x-a)$ and substituting for $x=a$ for finding the value polynomial gives you the same answer. I'm sure you must have proved the remainder theorem earlier:
$$f(x)=q(x)d(x)+r(x)$$
Where $q(x)\ \&\ r(x)$ are the quotient and remainder respectively obtained when $f(x)$ is divided by divisor $d(x)$ (knowing here that $\deg r(x) < \deg d(x)$ and $d(x)$ is non-zero polynomial). So if $f(x)$ is to be divided by some $x-a$ we get:
$$f(x)=(x-a)q(x)+r(x)$$
And it is not difficult to conclude that for the case $x=a$ we get:
$$f(a)=(a-a)q(a)+r(a)=0 \cdot q(a)+r(a)=0+r(a)=r(a)$$
It is important, and to maintain standards of rigor, that we have assumed at the beginning of the proof that $q(x)$ and $r(x)$ are obtained uniquely for division of $f(x)$ by $d(x)$, which is actually an assertion of Euclid's division algorithm. So I don't see why you wouldn't understand why you get the result.
Perhaps, you could read about Hornet's synthetic division. It is nothing but an optimized version in the case of divisor being a linear polynomial of the standard long division algorithm we use. You can check this video and this one.
