Stationarity of AR(2) process

Question

I am new to time series modeling and currently struggling with stationarity. Can someone please explain why the roots of the following AR polynomial are $ - 1 $ and $1/2$?

The AR(2) process is $X_t = X_{t-1} + 2 X_{t-2} + Z_t$

To my best knowledge so far, I could use backward shift operator writing

$(1- B + 2B^2)X_t = Z_t$

But I don`t know how to proceed with that.

Thank you in advance

Answer 1

The shift operator $1-B-2B^2$ factorizes as such: $$ 1-B-2B^2=(1+B)(1-2B). $$

Note that in your question you had a sign error when writing down the shift operator - the $2X_{t-2}$ moves from the right to the left side of the equation and thus becomes a $-2B^2$.