43. (ROM 2) The equation 
$$ x^{5}+5 \lambda x^{4}-x^{3}+(\lambda \alpha-4) x^{2}-(8 \lambda+3) x+\lambda \alpha-2=0 $$
is given. (a) Determine $\alpha$ such that the given equation has exactly one root independent of $\lambda$. (b) Determine $\alpha$ such that the given equation has exactly two roots independent of $\lambda$.
43. We can write the given equation in the form 
$$ x^{5}-x^{3}-4 x^{2}-3 x-2+\lambda\left(5 x^{4}+\alpha x^{2}-8 x+\alpha\right)=0 . $$ 
A root of this equation is independent of $\lambda$ if and only if it is a common root of the equations 
$$ x^{5}-x^{3}-4 x^{2}-3 x-2=0 \quad \text { and } \quad 5 x^{4}+\alpha x^{2}-8 x+\alpha=0 . $$ 
The first of these two equations is equivalent to $(x-2)\left(x^{2}+x+1\right)^{2}=0$ and has three different roots: $x_{1}=2, x_{2,3}=(-1 \pm i \sqrt{3}) / 2$. 
(a) For $\alpha=-64 / 5, x_{1}=2$ is the unique root independent of $\lambda$. 
(b) For $\alpha=-3$ there are two roots independent of $\lambda$ : $x_{1}=\omega$ and $x_{2}=\omega^{2}$.