Multiplicity of a root

Determine the multiplicity

Variant 1
of the root 1 in the polynomial \(x^4+2x^3+x^2+3x+3\) over \(\mathbb Z_5\).
Resolution
Divide the given polynomial by the term \((x-1)\) as long as the remainder is zero.
Result
\(x^4+2x^3+x^2+3x+3=(x-1)^2(x^2+4x+3)\), hence the multiplicity of 1 is 2.
Variant 2
the root 6 in the polynomial \(x^4+3x^3+6x^2+3x+6\) over \(\mathbb Z_7\).
Result
\(x^4+3x^3+6x^2+3x+6=(x-6)(x^3+2x^2+4x+6)\), the multiplicity is 1.
Variant 3
the root 1 in the polynomial \(x^4+x^3+2x+2\) over \(\mathbb Z_3\).
Result
\(x^4+x^3+2x+2=(x-1)^3(x+1)\), the multiplicity is 3.