Answer
$\{a_1x+a_2x^2+a_3x^3:\text{$a_1,a_2,a_3$ are real}\}$.
Work Step by Step
The kernel of a linear map $T : A → B$, is the set of all elements $v$ in $A$ for which $T(v) = 0$, where $0$ is the zero vector in B.
Hence, here we need $a_0=$. Thus $a_0=0$ and the others are free variables. Thus the kernel is $\{a_1x+a_2x^2+a_3x^3:\text{$a_1,a_2,a_3$ are real}\}$.
