Past exam of the algebra, analysis and differential equations 1 course of the University of Oxford / 2006 / 3 / ii / Solution

It is not an ideal. The constant polynomial one belongs to $J_2$, but its product with $X$ does not: the derivative of $X$ at zero is one. Thus $J_2$ fails closure under multiplication by arbitrary elements of $\mathbb{R}[X]$, a required ideal property. Closure under addition alone would not suffice.