<s>[INST] How do I prove that $x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+\dotsb+xy^{n-2}+y^{n-1})$ I am unsuccessfully attempting a problem from Spivak's popular book 'Calculus' 3rd edition. The problem requires proof for the following equation: 
$$x^n-y^n=(x-y)(x^{n-1}+x^{n-2}+\dotsb+xy^{n-2}+y^{n-1})$$
The solution to the problem, contained in the answer book, is as follows:
\begin{align*}
x^n-y^n &= (x-y)(x^{n-1}+x^{n-2}y+\dotsb+xy^{n-2}+y^{n-1})\\
&= x(x^{n-1}+x^{n-2}y+\dotsb+xy^{n-2}+y^{n-1})\\
&\qquad -[y(x^{n-1}+x^{n-2}y+{...}+xy^{n-2}+y^{n-1})]&\Rightarrow \mathbf{Equation1}\\
&=x^n+x^{n-1}y+\dotsb+x^2y^{n-2}+xy^{n-1}\\
&\qquad -[x^{n-1}y+x^{n-2}y^2+xy^{n-1}+y^n]&\Rightarrow \mathbf{Equation2}\\
&=x^n-y^n
\end{align*}
While I believe that the distributive law was used to arrive at Equation 1, I do not understand how Equation 2 was arrived at. 
I have tried to solve this independently to no avail. I cannot seem to understand how $x^n$ and $y^n$ came about in Equation 2 for example.
To summarise the question, what principle was Equation 2 based upon? And how was this applied in the above problem. 
 [/INST] The simplest way is to prove first
$$1-t^n=(1-t)(1+t+\dots+t^{n-1})$$
by induction.
The formula is trivial for $n=1$. So suppose the formula is valid for some $n\ge 1$, and consider the exponent $n+1$.
\begin{align}
1-t^{n+1}&=(1-t^n)+(t^n-t^{n+1})\\
&=(1-t)(1+t+\dots+t^{n-1})+t^n(1-t)\\
&=(1-t)(1+t+\dots+t^{n-1+t^n}),
\end{align}
 which proves the inductive step.
Now for the general formula: set $y=tx$, and substitute in the expression:
\begin{align}
x^n-y^n&=x^n(1-t^n)=x^n(1-t)(1+t+\dots+t^{n-1})\\
&=x(1-t)\cdot x^{n-1}(1+t+\dots+t^{n-1})\\
&=(x-tx)(x^{n-1}+x^{n-2}tx+\dots +(tx)^{n-1})\\
&=(x-y)(x^{n-1}+x^{n-2}y+\dots +y^{n-1}).
\end{align}
</s>