# Interactions terms and higher order polynomials

If I were interested in fitting two-way interactions between a linear explanatory variable $a$ and another explanatory variable $b$ that has a quadratic relationship with the dependent variable $y$, would I have to include both the interaction with the quadratic component and the interaction with the linear component in the model? E.g.: $$y\sim a+b+b^2+ab+ab^2$$ In turn building on my previous thread: Curvature terms and model selection, if this were a model selection analysis using MuMIn in R, with many explanatory variables, would the outputted models containing an interaction term that included a quadratic term $a:b^2$ only be valid if the interaction term with the linear component $a:b$ was also present in that same model, as well as $a$, $b$ and $b^2$ as direct effects?