If $\frac{1+\mathrm{i}\sqrt{3}}{2}$ is a root of the equation ${\mathrm{x}}^4 – {\mathrm{x}}^3 + \mathrm{x} – 1 = 0$ then its real roots are

$\begin{array}{l}Letf\left(x\right)=x^4-x^3+x-1=0\\ If\frac{1+i\sqrt{3}}{2}isarootthen\frac{1-i\sqrt{3}}{2}\\ TheEquationis{x}^2-(\frac{1+i\sqrt{3}}{2}+\frac{1-i\sqrt{3}}{2})x+\left(\frac{1+i\sqrt{3}}{2}\right)\left(\frac{1-i\sqrt{3}}{2}\right)=0\\ \Rightarrow x^2-x+1=0\\ Bysyntheticdivision\end{array}$

$\begin{array}{l}\therefore f\left(x\right)=(x^2-x+1)(x^2-1)=0\\ \therefore x^2-1=0\Rightarrow x=\pm 1\end{array}$