Let $R$ be a commutative ring. Let $p(x) = a_n x^n + a_{n-1} x^{n-1} +...+a_1 x +a_0 \in R[x]$.

Prove that, $p(x)$ is a unit in $R[x]$ iff $a_0$ is a unit and $a_1 , a_2 ,... , a_n $ are nilpotent in $R$

I could prove that $a_0$ is a unit $R$

and that $a_i $ is a zero divisor in $R$

but I couldn't show that it's a nilpotent !

any hints ?!

This is problem $\#33.(a)$ page 250 from Dummit and Foote's text , 3rd ed .

On a website, they said that the proof can be done by induction but I couldn't follow the proof.

any nice hints to help me solve this problem ?