6.353 Find all values of $\lambda$ for which the equations

$\lambda x^{3}-x^{2}-x+(\lambda+1)=0$ and $\lambda x^{2}-x-(\lambda+1)=0$ have a common root, and find this root.
Solution. Let $x_{0}$ be a common root,

$$
f(x)=\lambda x^{3}-x^{2}-x+(\lambda+1), g(x)=\lambda x^{2}-x-(\lambda+1)
$$

Represent $f(x)$ in the form (3). Then $r_{1}(x)=\lambda x+(\lambda+1) \Rightarrow r_{1}(x)=0$ when $x_{0}=-\frac{\lambda+1}{\lambda}$. But then $g\left(-\frac{\lambda+1}{\lambda}\right)=0$, from which $\frac{2(\lambda+1)}{\lambda}=0 \Leftrightarrow \lambda=-1, x_{0}=0$.

Answer: 0 when $\lambda=-1$.