TITLE: Find the remainder when $f(x)$ is divided by $(x - 3)$ given $f(1)$ and $f(7)$
QUESTION [8 upvotes]: A polynomial $f(x)$ is given. All we know about it is that all its coefficients are non-negative integers, $f(1) = 6$ and $f(7) = 3438$. Hence find the remainder when $f(x)$ is divided by $(x-3)$. I have no clue on how to go about solving this.

REPLY [13 votes]: The polynomial would be $x^4+ 3x^3 +x+1$.
The first observation $f(1)=6$ is that all the coefficients are less than $6$.
So $f(7)=3438$ is the decimal equivalent of the base 7 number formed by the coefficients $1 3 0 1 1$ ($=3438_{10}$).
So by remainder theorem the remainder is $f(3)= 166$.