Let the model $$y_t=\beta_0+\beta_1t+z_t\qquad t=1,2,\dots$$ $$z_t=\epsilon_t+\theta\epsilon_{t-1}$$ where $\epsilon_t$ is White noise with zero mean and variance $\sigma^2$ and $\beta_0,\beta_1,\theta$ constants.
a) Is $y_t$ stationary?
b) Is $(1-B)y_t$ stationary? (where $B$ is a lag operator)
This model can be written as $$y_t=\delta_t+\epsilon_t+\theta\epsilon_{t-1}$$
a) It is not a stationary process, this process have trend and clearly this have no constant mean, because $E[y_t]$ varies according to $t$.
b) $$(1-B)y_t=\beta_0+\beta_1t+\epsilon_t+\theta\epsilon_{t-1}-\beta_0-\beta_1(t-1)+\epsilon_{t-1}-\theta\epsilon_{t-2}$$ $$=\beta_1+\epsilon_t+\epsilon_{t-1}(\theta-1)+\theta\epsilon_{t-2}$$ $$=\beta_1+\epsilon_t[1+B(\theta-1)-\theta B^2]$$
$$B=\frac{(1-\theta)\pm \sqrt{(\theta-1)^2+4\theta}}{2}$$
The roots are $1$ and $-\theta$, so this process is not stationary?
I have other doubts too:
1) The process $y_t$ is a $MA(1)$ with non-zero mean?
2) When I look the roots of polynomial, both roots need to be $>1$ in modulus or just one is enough?