Find the length of the impulse response for the given output and input Homework Question:

Consider a signal $x[n]=\alpha e^{j \omega_{0} n}+\beta e^{j \omega_{1} n}+\gamma e^{j \omega_{2} n} .$ What is the length of impulse
response $h[n]$ of a system (non-trivial) such that $x[n] * h[n]=0$ ?

I have written the form in frequency domain in terms of system function as $$H(\omega)\big(\alpha \delta\left(w-w_{0}\right)+\beta \delta\left(w-w_{1}\right)+\gamma \delta\left(w-w_{2}\right)\big)=0$$
Thus $H(\omega)$ must have zeroes at $\omega$ = $\omega_0,\omega_1,\omega_2$
I am not sure about the meaning of impulse response length and also how to proceed to find the same from here.
You correctly figured out that the system must have $3$ zeros. So you just have to define a corresponding transfer function $H(z)$ that satisfies
$$H(z_i)=0,\qquad z_i=e^{j\omega_i},\quad i\in\{0,1,2\}$$
In other words, what is the minimum number of coefficients of a polynomial with $3$ zeros? Note that you might need to provide two solutions: one general solution, and one with the restriction that the polynomial coefficients be real-valued (as is required for a real-valued system).