What is the meaning of "due to the symmetry of the coefficients, if $x=r$ is a zero of $x^4+x^3+x^2+x+1$ then $x=\frac1r$ is also a zero" I was studying this answer about factoring $x^4+x^3+x^2+x+1$:
https://socratic.org/questions/how-do-you-factor-x-4-x-3-x-2-x-1
The author says: "A cleaner algebraic approach is to notice that due to the symmetry of the coefficients, if $x=r$ is a zero of $x^4+x^3+x^2+x+1$ then $x= {1\over r}$ is also a zero"
And eventually he writes  $x^4+x^3+x^2+x+1=(x^2+ax+1)(x^2+bx+1)$
Question $1$: What is the meaning of symmetry of the coefficients?
Question $2$: Can we do the same approach for $x^4-x^3+x^2-x+1$? ( I ask because it is relevant to my other question: Problem with factoring $x^4-x^3+x^2-x+1$)

To answer the original question, the thinking process comes as follows:
(1) If $r$ is a solution to $x^4-x^3+x^2-x+1=0$, then $r^4-r^3+r^2-r+1=0$.
(2) Divide both sides by $r^4$ you get $({1\over r})^4-({1\over r})^3+({1\over r})^2-({1\over r})+1=0$. Therefore $1\over r$ is also a solution.
(3) Hence if $(x-r)$ is a factor of the polynomial then $(x-{1\over r})$ is also a factor.
(4) Therefore the equation can be written as $(x-r)(x-{1\over r})(x-s)(x-{1\over s})$
(5) Therefore it can be written as $(x+ax+1)(x+bx+1)$
