Do data imputation and normalization when using polynomial regression

Question

The question is about the practical use of polynomial regression.
Let's say there is a dataset with columns A, B, T where T is a dependent variable, A and B are independent variables. A and B contain missing values. I want to fill in the gaps with the mean, then normalize values by the formula:

(x – u) / s,

where u is the mean and s is the standard deviation.
Everything is clear when I use linear regression. What about polynomial?
A^2, B^2 and AB columns are added for a quadratic polynomial case. How to fill AB, if the values ​​of A and B are missing?
By the product of averages? When calculating AB, should I multiply the normalized values ​​or should I normalize the result after?

Answer 1

I want to fill in the gaps with the mean, then normalize values

First, single imputations of missing predictor values are likely to lead to bias. See van Buuren's Flexible Imputation of Missing Data.

Second, there is usually no need to normalize the predictor values in this type of regression.

Third, for derived variables like $A^2$, $B^2$ and $AB$, van Buuren says in section 6.4.1:

The easiest way to deal with the problem is to leave any derived data outside the imputation process.

So your best choice is to do multiple imputation of the missing data on $A$ and $B$ and then just let standard design-matrix calculations produce the polynomial terms from the $A$ and $B$ values in each imputed data set.