Let $p(x) = a_0 + a_1 x + \ldots + a_n x^n$ be a non-zero polynomial with integer coefficients. If $p(\sqrt{2} + \sqrt{3} + \sqrt{6}) = 0$,then the smallest possible value of $n$ is

Convert the complex number $z = \frac{i-1}{\cos \frac{\pi}{3} + i \sin \frac{\pi}{3}}$ into polar form.

In a cubic equation,the coefficient of $x^2$ is $0$ and the remaining coefficients are real. If one root is $\alpha = 3 + 4i$,and the remaining roots are $\beta$ and $\gamma$,then find the value of $\alpha \beta \gamma$.

The complex numbers $\sin x + i \cos 2x$ and $\cos x - i \sin 2x$ are conjugate to each other,for

Let the product of $\omega_1=(8+i) \sin \theta+(7+4 i) \cos \theta$ and $\omega_2=(1+8 i ) \sin \theta+(4+7 i ) \cos \theta$ be $\alpha+ i \beta$,where $i =\sqrt{-1}$. Let $p$ and $q$ be the maximum and the minimum values of $\alpha+\beta$ respectively. Then the value of $p+q$ is equal to

If $z=3+5i$,then $z^3+\bar{z}+198$ is equal to