Polynomial regression for single feature data

Hello, I am a bit confused about using polynomial regression for single feature data.

Let’s say that I have a single feature data which can be modeled as F(x) = x+x^(2) (this is x squared not superscript)
will I be using two different weights w1 and w2? or just a single weight since i have only one feature.
Secondly, if i have to use two different weights w1 and w2, for one feature, can i use the numpy dot product to get w1x+w2x^(2) in my code?

sorry if the question seems kinda odd or is wrong, but i am really confused.

Every feature has its own weight.