Hello and thank you for taking interest in my question. This question came about when I was trying to integrate $$\int\frac{4x^5-1}{(x^5+x+1)^2} \,\mathrm{d}x$$ by the partial fraction method. I have found the solution by other methods and I have also seen the post in this site regarding this integral. I can't help but wonder if this can actually be integrated by using partial fractions.
So I tried to factor the denominator $x^5+x+1$ and found it to be $(x^3-x^2+1)(x^2+x+1)$. I am trying to rewrite $x^3-x^2+1$ as a product of a polynomial of degree 2 and a polynomial of degree $1$ but I cannot seem to calculate the root of $x^3-x^2+1$ i.e. I cannot find the solution of $x^3-x^2+1=0$. I graphed out $x^3-x^2+1$ and see it does have a root at $x=-0.755$. This may be stupid but may I please ask for help in algebraically finding the zero of this equation?
Thank You