Consider a specific instance of the difference equation (5.33): (left(1-a_1 mathrm{~B}-cdots-a_p mathrm{~B}^p ight)left(f_t ight)-a_0=0 . tag{5.33})

Consider a specific instance of the difference equation (5.33):

   \(\left(1-a_1 \mathrm{~B}-\cdots-a_p \mathrm{~B}^p\right)\left(f_t\right)-a_0=0 . \tag{5.33}\)

\[ \left(1-3 \mathrm{~B}+2 \mathrm{~B}^{2}\right)\left(x_{t}\right)=0, \quad \text { for } t=2,3, \ldots \]

(a) What is the characteristic polynomial for this difference equation, in the form of equation (5.35) (that is, with \(x_{t}=z^{-t}\) )? What are its roots?

(b) What are the general solutions to this difference equation?

Show that your solutions are indeed solutions to the difference equation.

\(1-a_1 z-\cdots-a_p z^p, \tag{5.35}\)