How to truncate a polynomial?

Given \(9 x^{5}+4 x^{4}+3 x^{3}+x^{2}+x +1\) how to truncate it, so that all terms of \(x^3\) and higher are removed?

This can be done as follows

restart; 
p:=1+x+x^2+3*x^3+4*x^4+9*x^5; 
simplify(p,{x^3=0})

\[ x^{2}+x +1 \]