Find Rational Roots and Factor $x^4 + x^3 - 4x^2 + x + 1$ Find all rational roots of the polynomial $x^4 + x^3 - 4x^2 + x + 1$. Factor this polynomial into a product of monic irreducible polynomials over the rationals. 
I am confused about this question, but isn't the rational root theorem tell us the roots are only $\pm1$?

\begin{align}
x^4+x^3-4x^2+x+1 & = (x^4 - 2x^2 + 1) + (x^3 - 2x^2 + x) = (x^2-1)^2 + x(x-1)^2\\
& = (x-1)^2 \left( (x+1)^2 + x \right) = (x-1)^2(x^2 + 3x +1)\\
& = (x-1)^2 \left( \left(x+\dfrac32\right)^2 + 1 - \dfrac94\right)\\
& = (x-1)^2 \left( \left(x+\dfrac32\right)^2 - \dfrac54\right) = (x-1)^2 \left( \left(x+\dfrac32\right)^2 - \left(\dfrac{\sqrt5}2 \right)^2\right)\\
& = (x-1)^2 \left(x + \dfrac{3+\sqrt5}2\right) \left(x + \dfrac{3-\sqrt5}2\right)
\end{align}
