Simplifying $1 - x + x^2 - x^3 + ... + x^{98} - x^{99}$ to an equivalent expression. I am doing an exercise to see the error when solving this polynomial for $x = 1.00001$ using nested multiplication.
I believe the correct way to achieve this simplification (based on a lecture) is to multiply the polynomial by $\frac{1+x}{1+x}$; however, my algebra skills are not really up to par so I am failing to see the purpose of this. I assume it is to cancel out most of the terms in the polynomial - but which ones are being canceled?

Here's the beginning:
$$\begin{align}
&\ \ \ 1-x+x^2-x^3+\cdots+ x^{98}-x^{99}\\
&= \left(1-x+x^2-x^3+\cdots+ x^{98}-x^{99}\right)\frac{1+x}{1+x}\\
&= \frac{1(1+x)-x(1+x)+x^2(1+x)+\cdots +x^{98}(1+x)-x^{99}(1+x)}{1+x}\\
&=\frac{1+x-x-x^2+x^2+x^3-x^3\cdots +x^{98}+ x^{99}-x^{99} -x^{100}}{1+x}\\
\end{align}$$
Can you see where this goes?
