Polynomial Division - Remainder when divisor is squared?? 
Let $f(x)$ be a polynomial. If $(x-a)$ is a factor of $f(x)$, prove that if $f(x)$ is divided by $(x-a)^2$, the remainder will be equal to $n(x-a)$ for some real value $n$.

I have tried setting $f(x) $ to equal $g(x)(x-a)^2+k(x-a)$, but all that really proves that an equation with this form will satisfy it. I have no idea where to go from here, and to say that any equation following this form will work seems really flawed. Any help is appreciated.
 A: Note that $f(x) = (x-a)g(x)$ for some $g(x)$ with degree $1$ less than $f(x)$. Also note that since $\deg(f)\geq 2, g(x)= (x-a)h(x) + r$ for some real $r$ and $h(x)$ with degree $2$ less than $f(x)$. Substituting back, we obtain:
$f(x) = h(x)(x - a)^{2} + r(x - a)$
Thus, $f(x)$'s remainder when divided by $(x-a)^{2}$ is of the form $r(x-a)$. $\blacksquare$
A: It follows from Euclidean division of polynomials (https://en.m.wikipedia.org/wiki/Polynomial_long_division) that
$f(x) = (x - a)^2g(x) + r(x), \tag 1$
where either
$r(x) = 0 \tag 2$
or
$\deg r(x) < 2; \tag 3$
in case (2), we take
$n = 0, \tag 4$
whence
$r(x) = 0 = n(x - a) = 0(x - a); \tag 3$
in case (3) we have
$\deg r(x) \le 1, \tag 4$
and thus $r(x)$ may be written
$r(x) = \alpha x + \beta, \; \alpha, \beta \in \Bbb R; \tag 5$
we evaluate (1) at
$x = a, \tag 6$
and obtain
$f(a) = (a - a)^2g(a) + r(a) = r(a); \tag 7$
since $x - a$ is a factor of $f(x)$,
$f(x) = (x - a)h(x), \tag 8$
for some
$h(x) \in \Bbb R[x]; \tag 9$
from (8),
$f(a) = (a - a)h(a) = 0, \tag{10}$
then (5), (7) and (10) together become
$r(a) = \alpha a + \beta = 0, \tag{11}$
so
$\beta = -\alpha a, \tag{12}$
and then
$r(x) = \alpha x - \alpha a = \alpha (x - a), \tag{13}$
which is of the requisite form with
$n = \alpha; \tag{14}$
that is,
$f(x) = (x - a)^2g(x) + \alpha(x - a). \tag{15}$
A: Let $f(x)= (x-a)^2g(x)+ r(x)$. Clearly $deg(r)\leq 1$ . Given $(x-a)$ divides $f(x)$. Therefore $(x-a)$ divides $(x-a)^2g(x)+ r(x)$. This implies $(x-a)$ divides $r(x)$. Therefore $r(x)= n(x-a)$ ( where $n$ is a real value ) as $deg(r) \leq 1$.
