Kernel of endomorphism for polynomials from $f(x)$ to $f(2x+1)$

Question

I am looking for a kernel of a map, producing $f(2x+1)$ out of $f(x)$. where $f(x)$ is an arbitrary polynomial of degree $n$. I thought of trying to write this transformation as a matrix and then to figure out the kernel, but I struggled already at this point at finding the matrix. Thanks for help

Answer 1

The kernel of the linear operator in question is the set of polynomials $p$ up to degree $n$ with $p(2x+1)=0$ identically. Comparing coefficients, starting with $x^n$ and working downwards, yields that all the coeffients of $p$ are zero, so the kernel consists only of the zero polynomial. (The map $x\mapsto2x+1$ is invertible, another way to see that the kernel is trivial.)