30. Find the largest positive number $x$ such that
$$
\left(2 x^{3}-x^{2}-x+1\right)^{1+\frac{1}{2 x+1}}=1
$$
30. Ans: 1
Since the exponent on the left side of the given equation is non-zero if $x>0$, it follows that the equation is possible only when
$$
2 x^{3}-x^{2}-x+1=1 \Longrightarrow 2 x^{3}-x^{2}-x=0 \Longrightarrow x(x-1)(2 x+1)=0 \Longrightarrow x=0,1,-\frac{1}{2} \text {. }
$$

Since $x>0$, it follows that $x=1$.