So I have found the following problem in my textbook without solutions, which presents the $AR(2)$ process defined by $$X_{t} = 0.5X_{t-1} + 0.25 X_{t-2} + Z_{t}$$ I am asked what the polynomial $\phi(z)$ for this model is, and further to write the polynomial as a product of linear terms in $z$ and show how to expand the inverse of each of these linear terms as a power series in $z$, while showing why these power series converge for small enough $z$.
The first part of the problem I found easy, and I found the polynomial is $$ \phi(B) = 1 - 0.5B - 0.25 B^{2}$$ with roots $r_{1}, r_{2} = \dfrac{0.5 \pm \sqrt{1.25}}{-0.5}$.
$\implies$ since the roots lie outside the unit circle, there exists a solution in the causal form $\sum \limits_{j=0}^{\infty} \theta_{j} Z_{t-j}$.
But I do not know what is meant by "expanding the inverse of each linear term as a power series", or how I should go about doing that for this problem. Further, I don't really see the purpose?
Informally, it is sufficient to find the power series representation of the rational function $$f(x) = \frac{1}{1 - 0.5x - 0.25x^2} = \frac{4}{r_2 - r_1}\left(\frac{1}{r_2}\frac{1}{1 - \frac{x}{r_2}} - \frac{1}{r_1}\frac{1}{1 - \frac{x}{r_1}}\right).$$ Now use the celebrated geometric sequence equality $$\frac{1}{1 - z} = \sum_{k = 0}^\infty z^k,$$ for all $z$ such that $|z| < 1$.