Degree of exponential polynomial

Question

What is the degree of the exponential polynomial :$$ e^x * (x^2 + x + 1)$$?

I am getting confused because of the exponential term there...

Answer 1

The idea is that one considers the vector space $V$ of all functions $f$ of the form $f(x)=e^x\,p(x)$, where $x\mapsto p(x)$ is a polynomial of degree $\leq 2$. This $V$ has dimension $3$. The operator $D:\>f\mapsto f'$ is linear, maps $V$ to itself, and has rank $3$, since $D(e^x\,x^k)=e^x q(x)$ is again in $V$, and $q$ is of degree $k$. It follows that $D:\>V\to V$ is nonsingular, in particular: surjective. This implies that any given function $f(x)=e^x(a_0+a_1x+a_2 x^2)\in V$ is the derivative of some function $F(x)=e^x(b_0+b_1x+b_2 x^2)\in V$. The $b_k$ can be determined by comparing coefficients of $F'$ and the given $f$.