Polynomial remainder problem. A polynomial $f(x)$ is such that upon division by $(x-3)$ it leaves a remainder of $15$ and upon division by $(x-1)^2$, it leaves a remainder of $2x+1$.
What is the remainder when $f(x)$ is divided by $(x-3)(x-1)^2$?

Hint. We have that
$$f(x)=A(x)(x-3)+15=B(x)(x-1)^2+2x+1=C(x)(x-3)(x-1)^2+r(x)$$
where $r(x)=ax^2+bx+c$ (because the degree of $(x-3)(x-1)^2$ is $3$) and $A,B,C$ are polynomials.
In order to find $a$, $b$, and $c$, let $x=1$ and $x=3$. 
Consider also the derivative of $f$ at $x=1$.
Therefore we obtain the equations:
$$r(3)=15,\quad r(1)=3,\quad r'(1)=2.$$
Are you able to find $a,b,c$ now?
