$($ii$)$Consider a characteristic polynomial $x^5+x+1.$ Now answer the following and you must show your work for full credit.

$(7)$

$($ii$.$a$)$Is this polynomial primitive?

$($ii$.$b$)$Give an internal exclusive$-$OR $($modular$)$ realization of this polynomial.

$($ii$.$c$)$Design a four stage cellular automata whose cells follow Rule $90.$