Finding primary decompositions of ideals

Question

I have been given this example of the decomposition of an ideal into primary ideals

$$ I =⟨x^2,xy,x^2z^2,yz^2⟩$$

Then the primary decomposition of this ideal is:

$$⟨x^2,y⟩∩⟨x,z^2⟩⊆K[x,y,z]$$

This makes sense but in general what are the techniques used to find primary decompositions as in my notes the definition of primary decomposition is given followed by a single example which doesn't really help me when generally finding decompositions

thanks for the help

Answer 1

$\langle x^2,\underline{xy},yz^2\rangle=\langle x^2,{\color{red}{x}},yz^2\rangle\cap \langle x^2,{\color{red}{y}},yz^2\rangle=\langle x,\underline{yz^2}\rangle\cap \langle x^2,y\rangle=\langle x,{\color{red}y}\rangle\cap\langle x,{\color{red}{z^2}}\rangle\cap \langle x^2,y\rangle=\langle x^2,y\rangle\cap\langle x,z^2\rangle$