Question 75
Let $a_{1}, a_{2}, \ldots, a_{n}, k$, and $M$ be positive integers such that
$$
\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}}=k \quad \text{ and } \quad a_{1} a_{2} \ldots a_{n}=M .
$$
If $M\gt 1$, prove that the polynomial
$$
P(x)=M(x+1)^{k}-\left(x+a_{1}\right)\left(x+a_{2}\right) \cdots\left(x+a_{n}\right)
$$
has no positive roots.
$$
\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}}=k \quad \text{ and } \quad a_{1} a_{2} \ldots a_{n}=M .
$$
If $M\gt 1$, prove that the polynomial
$$
P(x)=M(x+1)^{k}-\left(x+a_{1}\right)\left(x+a_{2}\right) \cdots\left(x+a_{n}\right)
$$
has no positive roots.