Question

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

• A. $1,1$
• B. $-1,-1$
• C. $1,2$
• D. $1,$ $-1$

Answer 1

The correct option is D $1,$ $-1$

Let $\alpha ,\beta ,\gamma ,\delta$ be the roots of $x^4-x^3+x+1=0$
Now let $\alpha =\frac{1+i\sqrt{3}}{2}$ and $\beta =\frac{1-i\sqrt{3}}{2}$
As complex roots exists in conjugate pair.
Now $\alpha +\beta +\gamma +\delta =1\Rightarrow 1+\gamma +\delta =1\Rightarrow \gamma +\delta =0$
Therefore, observing options
A) $\gamma +\delta =1+1=2$
B) $\gamma +\delta =-1-1=-2$
C) $\gamma +\delta =1+2=3$
D) $\gamma +\delta =1-1=0$
Hence, option D is correct answer.