OIM 1988 Problem 5

Consider all the numbers of the form $x+yt+zt^2$ , with $x,y,z$ rational numbers and $t=\sqrt[3]{2}$ . Prove that if $x+yt+zt^2\not= 0$ , then there exist rational numbers $u,v,w$ such that $$(x+yt+z^2)(u+vt+wt^2)=1$$