1-x-x^2

The inverse of 1-x² can be calcuated two ways.

First way: We know 1/(1-y)=(1+y+y²+y³+...) so substitute y=2x, and it comes out.

Second way: Do it directly.

1. The constant coefficent must be 1.
2. The x coefficient must 2.
3. The x² coefficient must 4.
4. The x³ coefficient must 8.

We did a couple more examples of computing an inverse. Then we went to an interesting one.

What is the multiplicative inverse of 1-x-x².

It turns out to be 1+x+2x²+3x³+5x⁴+8x⁵+13x⁶...

The coefficients are the fibonacci series!