<s>[INST] 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?!
 [/INST] Let $f(x) = x^5 + 5x^4 + 1.$
Rather than perform division of polynomials, a simple way to test possible rational roots is just to plug them into the equation. For $x=-1$ and $x=1$ this gives us
\begin{align}
f(-1) &= (-1)^5-5(-1)^4 + 1 = -5, \\
f(1)  &= (1)^5-5(1)^4 + 1 = -3.
\end{align}
But of course $f(0) = 1.$ So there is one root between $-1$ and $0$ and another between $0$ and $1.$ There must also be a root greater than $1$ since  we know $f(x)$ eventually grows without bound.
That's three distinct roots. If any of these had multiplicity $4,$
then $f(x)$ would have a total of at least six roots (including multiple roots).
But $f(x)$ is only fifth degree and can have at most five roots.
Therefore $f(x)$ has no roots of multiplicity $4.$
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