Question 1
Let $P(x)=x^4-x^3+x$.
a) Prove that for every positive real $a$, the polynomial $P(x)-a$ has exactly one positive root.
b) Define $a_1=1/3$ and, for $n\ge1$, let $a_{n+1}$ be the positive root of $P(x)-a_n$. Prove that $(a_n)$ has a finite limit and find it.
a) Prove that for every positive real $a$, the polynomial $P(x)-a$ has exactly one positive root.
b) Define $a_1=1/3$ and, for $n\ge1$, let $a_{n+1}$ be the positive root of $P(x)-a_n$. Prove that $(a_n)$ has a finite limit and find it.