Show that $x^{21}+2x^8+1$ does not have multiple zeroes in any extension of $Z_3$.
Try
Here we have to show that $f(x) $ and $f'(x) $ have no common factor.
Now $f(x) =x^{21}+2x^8+1$
and $f'(x) =21x^{20}+16x^7=x^7$
Now the factors of $x^7$ are $x, x^2,...,x^7$ but none of these divides $f(x) $. Hence they share no common factor.
Am I correct? Also want to know any alternative approach!
That's all there is to it.
An alternative could be to be completely factor this polynomial. There are algorithms for that but I am reluctant to try that with pencil & paper alone (unless I spot a trick). A CAS will quickly tell you that this polynomial is the product of two irreducible factors. One of degree 7 and the other of degree 14.