Suppose that 3 is a unit.
Then 3 * p(x) = 1 for some p(x) in Z[x].
By comparing degrees,
deg (3 p(x)) = deg 3 + deg p(x) = deg 1
==> 0 + deg p(x) = 0
==> deg p(x) = 0.
However, that would imply that p(x) = 1/3, which is not an integer.
Hence, 3 is not a unit in Z[x].
I hope this helps!
Then 3 * p(x) = 1 for some p(x) in Z[x].
By comparing degrees,
deg (3 p(x)) = deg 3 + deg p(x) = deg 1
==> 0 + deg p(x) = 0
==> deg p(x) = 0.
However, that would imply that p(x) = 1/3, which is not an integer.
Hence, 3 is not a unit in Z[x].
I hope this helps!