Common factors in ARIMA(p,d,q)

Question

I have some concerns regarding interpreting ARIMA processes, A general ARIMA process is on the form $$ \phi(B)X_t = \theta(B)Z_t,\,\,Z_t\sim WN(0,\sigma^2)$$

For example if I have $$Y_t = (1-B)^{12}X_t$$ and \begin{align*} (5 + 6B + B^2)Y_t & = (5+B)Z_t \\ (5+B)(1+B)Y_t & = (5+B)Z_t \end{align*} Can one cancel out $5 + B$ ? so that $Y$ becomes a ARMA(1,0) process which is invertible but not causal ? $$(1+B)Y_t=Z_t$$ and $X$ will be a ARIMA$(1,12,0)$

Answer 1

Yes, but this follows from a general property of backshift operators provided (in your case) that $(1+B)Y_1=Y_1+Y_0=Z_1$.