Question

The common roots of the equation $x^3+2x^2+2x+1=0$ and $x^{1988}+x^{130}+1=0$ are (where $\omega$ is a non real cube root of unity).

• A. $\omega$
• B. ${\omega }^2$
• C. $-1$
• D. $\omega -{\omega }^2$

Answer 1

Answer: (A), (B)

The correct options are
A $\omega$
B ${\omega }^2$

$x^3+2x^2+2x+1=0$
$(x^3+1)+2x(x+1)=0$
$(x+1)(x^2-x+1)+2x(x+1)=0$
$(x+1)(x^2+x+1)=0$
$x=-1$ and $x^2+x+1=0$ implies $x=w,w^2$ where $w$ and $w^2$ are cube roots of unity.
Now $x=-1$ does not satisfy the second equation,
Hence the common roots are $w,w^2$,

since, $w^{1988}+w^{130}++1=1.w^2+1.w+1=0$

and $w^{2\times 1988}+w^{2\times 130}++1=1.w+1.w^2+1=0$. since, ($w^3=1$.)