How one can show that this seven degree polynomial has no real roots in $(0,1)$?

Question

How one can show that this seven degree polynomial has no real roots in the interval $(0,1)$: $$z^7+20z^6-z^5-576=0.$$ I have no idea to start

Answer 1

Over the interval $(0,1)$ the first two terms of the polynomial $z^7 + 20 z^6$ assume values in the interval $(0,21)$. Adding the constant term $-576$ to any value in this interval always yields a negative number. And all of the other terms also assume negative values over $(0,1)$. Therefore the polynomial as a whole assumes only negative values over $(0,1)$. As others said, without a sign change in the interval, there is no root.

Answer 2

It's easy to show that $f$ is increasing on $[0,1]$; furthermore $f(0)=-576$ and $f(1)=-556$.