Simulate data such that specific condition is met in polynomial regression

Questions
1

Hello,
I want to create a dataset (e.g., N = 20) based on the following polynomial regression:

Screenshot2022-10-06at15.36.12.png

The only condition that should be met is y= 0 for x = 1, 3, or 7. I don’t care about the epsilons for the moment, but I would be very happy to find a solution on how to generate the corresponding x values.
Thank you!

2

How about this:

Table[RandomChoice[{x - 1, x - 3, 
     x - 7}] (RandomInteger[{-3, 3}] x^2 + RandomInteger[{-3, 3}] x + 
     RandomInteger[{-3, 3}]) // Expand, {20}]