Factoring $x^5 - 5x^4 + 1$ The original problem was stated like this:
Prove that the polynomial $x^5 - 5x^4 + 1$ does not have roots of multiplicity 4.
So, factoring the polynomial would answer the question, but I don't know how to do it.
Using the root test, ${1, -1}$ should be roots, but If I divide by $x-1$ or $x+1$  $\frac {x^5 - 5x^4 + 1}{x-1}$, or $\frac {x^5 - 5x^4 + 1}{x-1}$ I get a remainder, so those are not roots?!

extended gcd in $\mathbb Q[x].$ the original polynomial and its derivative are coprime; there are no multiple roots, not even double.
$$  \left(   x^{5}  - 5 x^{4}  + 1 \right)  $$ 
$$  \left(   x^{4}  - 4 x^{3}  \right)  $$ 
$$  \left(   x^{5}  - 5 x^{4}  + 1 \right)  =  \left(   x^{4}  - 4 x^{3}  \right)  \cdot \color{magenta}{ \left(   x  - 1 \right)}  +  \left(   - 4 x^{3}  + 1 \right)  $$ 
 $$  \left(   x^{4}  - 4 x^{3}  \right)  =  \left(   - 4 x^{3}  + 1 \right)  \cdot  \color{magenta}{ \left(   \frac{  -  x  + 4 }{ 4 }  \right)}  +  \left(   \frac{  x  - 4 }{ 4 }  \right)  $$ 
 $$  \left(   - 4 x^{3}  + 1 \right)  =  \left(   \frac{  x  - 4 }{ 4 }  \right)  \cdot  \color{magenta}{ \left(   - 16 x^{2}  - 64 x  - 256 \right)}  +  \left( -255  \right)  $$ 
 $$  \left(   \frac{  x  - 4 }{ 4 }  \right)  =  \left( -255  \right)  \cdot  \color{magenta}{ \left(   \frac{  -  x  + 4 }{ 1020 }  \right)}  +  \left( 0 \right)  $$ 
 $$ \frac{ 0}{1} $$ 
 $$ \frac{ 1}{0} $$ 
 $$  \color{magenta}{ \left(   x  - 1 \right) }  \Longrightarrow  \Longrightarrow  \frac{  \left(   x  - 1 \right) }{ \left( 1  \right) } $$ 
 $$   \color{magenta}{\left(   \frac{  -  x  + 4 }{ 4 }  \right)}   \Longrightarrow  \Longrightarrow  \frac{  \left(   \frac{  -  x^{2}  + 5 x  }{ 4 }  \right) }{ \left(   \frac{  -  x  + 4 }{ 4 }  \right) } $$ 
 $$   \color{magenta}{\left(   - 16 x^{2}  - 64 x  - 256 \right)}   \Longrightarrow  \Longrightarrow  \frac{  \left(  4 x^{4}  - 4 x^{3}  - 16 x^{2}  - 319 x  - 1 \right) }{ \left(  4 x^{3}  - 255 \right) } $$ 
 $$  \color{magenta}{ \left(   \frac{  -  x  + 4 }{ 1020 }  \right)}   \Longrightarrow  \Longrightarrow  \frac{  \left(   \frac{  -  x^{5}  + 5 x^{4}  - 1 }{ 255 }  \right) }{ \left(   \frac{  -  x^{4}  + 4 x^{3}  }{ 255 }  \right) } $$ 
 $$  \left(   x^{5}  - 5 x^{4}  + 1 \right)  \left(   \frac{ 4 x^{3}  - 255 }{ 255 }  \right)  -  \left(   x^{4}  - 4 x^{3}  \right)  \left(   \frac{ 4 x^{4}  - 4 x^{3}  - 16 x^{2}  - 319 x  - 1 }{ 255 }  \right)  =  \left( -1  \right)  $$ 
