Find the factorization of the polynomial $x^5-x^4+8x^3-8x^2+16x-16$ as a product of irreducible on rings $R[x]$ and $C[x]$
Testing with the simplest possible root in this case, $P(1)=0$
Applying the schema of Ruffini
$ \begin{array}{c|lcr} & 1 & -1 & 8 & -8 & 16 & -16 \\ 1 & & 1 & 0 & 8 & 0 & 16 \\ \hline & 1 & 0 & 8 &0 &16 & 0 \\ \end{array} $
$P(x)=(x-1).(x^4+8x^2+16)$
$P(x)=(x-1).(x^2+4)^2 \rightarrow Irreducible$ $in$ $R[x]$
So, I must to factorize this to make it irreducible in $C[x]$
Applying Bhaskara in $(x^2+4)^2$ $a=1; b=0; c=4$
$$\frac{\pm\sqrt{-4.1.4}}{2}=\frac{\pm\sqrt{-16}}{2}= R_1, R_2$$ $R_1=-2i$
$R_2=2i$
So (...)
$P(x)= (x-2i).(x+2i).(x-1) \rightarrow Irreducible $ $in $ $C[x]$
this is well done?