Expanding brackets problem: $(z - 1)(1 + z + z^2 + z^3)$ I have:
$(z - 1)(1 + z + z^2 + z^3)$
As, I have tried my own methods and enlisted the help on online software, but as well as them not all arriving at the same solution, I can't follow their reasoning.
I tried to gather all the like terms:
$(z - 1)(z^6+1)$
And then thought it may offer a route to a difference or two cubes if I shifted powers across the brackets, but that's $z^7$ in total, isn't it? I can't just do $z^7-1$, can I?
If I try to expand, I see:
$(z - 1)(z^6 + 1) = z^7+z-z^6-1$
Well...
$z^7-z^6=z$
Add the other z leaves:
$z^2-1$?
I expect this is all wrong.
Will you help please?
On a side note, I have difficulty with understanding how I'm expected to tag this post accurately when I can't used "expand", "multiply", or use any of the subjects it covers as tags? 

I like to do factorizations like this by writing 
$$
z^7-1=(z-1)(\cdots)
$$
and try to figure out what should replace the dots.  First, we'd like to have a factor of $z^7$ in the end result, so the $z$ on the right side should be multiplied by $z^6$ to get this.  Then, we write
$$
z^7-1=(z-1)(z^6+\cdots).
$$
At this point, by the distributive law, the right side will multiply out to $z^7-z^6+\cdots$.  Since we don't want the $z^6$ in the final product, it must cancel with something.  We can cancel the $-z^6$ by multiplying $z$ by $z^5$ to get
$$
z^7-1=(z-1)(z^6+z^5+\cdots)
$$
because after distributing, the product is $z^7-z^6+z^6-z^5+\cdots$ and the $z^6$'s cancel.  Now, we're left with $z^7-z^5+\cdots$, so the $-z^5$ must cancel with another term, since $z^5=z\cdot z^4$, we have
$$
z^7-1=(z-1)(z^6+z^5+z^4+\cdots).
$$
At this point, the product is $z^7-z^4+\cdots$.  In order to cancel the $-z^4$, we introduce a $z^3$, which results in an extra $-z^3$.  This extra term is cancelled with a $z^2$, which results in an extra $-z^2$ in the product.  This $-z^2$ is cancelled with a $z$, but that gives an extra $-z$ in the product.  The extra $-z$ is cancelled with a $-1$.  Therefore, we have
$$
z^7=(z-1)(z^6+z^5+z^4+z^3+z^2+z^1+1+\cdots).
$$
But, after multiplying everything out, we find that the left side already matches the right side and there is nothing more in the dots!  Therefore,
$$
z^7=(z-1)(z^6+z^5+z^4+z^3+z^2+z^1+1).
$$
