I am reading the "An introduction to quantum computing for non-physicists" by Rieffel and Polak. In page 308, they define an entangled state as a state that "cannot be described in terms of the state of each of its components." They used the following example, $$ \frac{1}{{\sqrt 2 }}\left| {\left. {00} \right\rangle } \right. + \frac{1}{{\sqrt 2 }}\left| {\left. {11} \right\rangle } \right. = \left( {a_1 \left| {\left. 0 \right\rangle } \right. + b_1 \left| {\left. 1 \right\rangle } \right.} \right) \otimes \left( {a_2 \left| {\left. 0 \right\rangle } \right. + b_2 \left| {\left. 1 \right\rangle } \right.} \right) $$
In this example, there are no valid values for $a_1, a_2, b_1, b_2.$ As a result, this state is entangled.
My question is: with the above definition, is the following state entangled? $$ \frac{{\sqrt 2 }}{4}\left| {\left. {000} \right\rangle } \right. + \frac{{\sqrt 2 }}{4}\left| {\left. {001} \right\rangle } \right. + \frac{{\sqrt 2 }}{4}\left| {\left. {010} \right\rangle } \right. + \frac{{\sqrt 2 }}{4}\left| {\left. {011} \right\rangle } \right. + \frac{{\sqrt 2 }}{4}\left| {\left. {100} \right\rangle } \right. + \frac{{\sqrt 2 }}{4}\left| {\left. {101} \right\rangle } \right. + \frac{{\sqrt 2 }}{4}\left| {\left. {110} \right\rangle } \right. + \frac{{\sqrt 2 }}{4}\left| {\left. {111} \right\rangle } \right. $$
Note that the above state can indeed be decomposed to $$ = \left( {\frac{1}{{\sqrt 2 }}\left| {\left. 0 \right\rangle } \right. + \frac{1}{{\sqrt 2 }}\left| {\left. 1 \right\rangle } \right.} \right) \otimes \left( {\frac{1}{{\sqrt 2 }}\left| {\left. 0 \right\rangle } \right. + \frac{1}{{\sqrt 2 }}\left| {\left. 1 \right\rangle } \right.} \right) \otimes \left( {\frac{1}{{\sqrt 2 }}\left| {\left. 0 \right\rangle } \right. + \frac{1}{{\sqrt 2 }}\left| {\left. 1 \right\rangle } \right.} \right) $$