Question

Find an odd prime p for which x - 2 is a divisor of${\mathit{x}}^{\mathbf{4}}\mathbf{+}{\mathit{x}}^{\mathbf{3}}\mathbf{+}\mathbf{3}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathit{x}\mathbf{+}\mathbf{1}$ in role="math" localid="1648624672660" ${\mathbf{\mathbb{Z}}}_{\mathbf{p}}\mathbf{\left[}\mathbf{x}\mathbf{\right]}$.

Short answer

In ${\mathbf{\mathbb{Z}}}_{\mathbf{3}}\left[x\right]$, it has ${\left(x-2\right)}^4=x^4+x^3+3x^2+x+1$.

Step-by-step solution

Determine the odd number

Consider the given equation, x - 2 is a divisor of$x^4+x^3+3x^2+x+1$ in ${\mathrm{\mathbb{Z}}}_p\left[x\right]$.

Remember that in ${\mathbf{\mathbb{Z}}}_{\mathbf{3}}\left[x\right]$, $x-2=x+1$then, it computes as:

${\left(x-2\right)}^4={\left(x+1\right)}^4=x^4+x^3+3x^2+x+1$.