\b[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.

Solution

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

 Solution (source code)