\[(y - 1)\left( y^{2} + y + 1 \right)\left( y^{6} + y^{3} + 1 \right) =\]
\[= \left( y^{3} + y^{2} + y - y^{2} - y - 1 \right)\left( y^{6} + y^{3} + 1 \right) =\]
\[= \left( y^{3} - 1 \right)\left( y^{6} + y^{3} + 1 \right) =\]
\[= y^{9} + y^{6} + y^{3} - y^{6} - y^{3} - 1 =\]
\[= y^{9} - 1.\]