<s>[INST] trying to derive expression with polynomial arithmetic I'm trying to figure out how this is derived in "Time Series Analysis - Forecasting and Control" (Box, Jenkins): 
$$
a_t = \frac{1}{1-\theta B}z_t = (1 + \theta B + \theta^2 B^2+ ... +  + \theta^k B^k)(1-\theta^{k+1} B^{k+1})^{-1}z_t
$$
I can derive this : 
$$\frac{1}{1-\theta B} = (1 + \theta B + \theta^2 B^2+ ... ) = (1 + \theta B + \theta^2 B^2 + ... + \theta^kB^k) + \frac{\theta^{k+1}B^{k+1}}{1-\theta B} $$
How does one derive the top expression? 
 [/INST] Here's the way I got through it:
$\frac{1}{1 - \theta B} = (1 + \theta B + \theta^2 B^2 + ...) = (1+ \theta B + \theta ^2B^2 + ... + \theta^kB^k) + (\theta^{k + 1}B^{k + 1} + \theta^{k + 2}B^{k + 2} + ...)$
$ = (1+ \theta B + \theta ^2B^2 + ... + \theta^kB^k) + \theta^{k+1}b^{k+1}(1 + \theta B + \theta^2 B^2 + ...)$
So now we have 
$(1 + \theta B + \theta^2 B^2 + ...) = (1+ \theta B + \theta ^2B^2 + ... + \theta^kB^k) + \theta^{k+1}b^{k+1}(1 + \theta B + \theta^2 B^2 + ...)$.
Subtracting gives
$(1 + \theta B + \theta^2 B^2 + ...) - \theta^{k+1}b^{k+1}(1 + \theta B + \theta^2 B^2 + ...) = (1+ \theta B + \theta ^2B^2 + ... + \theta^kB^k)$
$\Rightarrow (1 - \theta^{k+1}b^{k+1})(1 + \theta B + \theta^2 B^2 + ...) = (1+ \theta B + \theta ^2B^2 + ... + \theta^kB^k)$
And finally, dividing both sides by $(1 - \theta^{k+1}b^{k+1})$, gives the desired result:
$\frac{1}{1 - \theta B} = (1+ \theta B + \theta ^2B^2 + ... + \theta^kB^k)(1 - \theta^{k+1}b^{k+1})^{-1}$
</s>