Let a$0,$ a$1,$ a$2,$ a$3,$ a$4$ be constant real numbers such that

a$0$

$5+$ a$1$

$4+$ a$2$

$3+$ a$3$

$2+$ a$4=0.$

Show that the polynomial P $($x$)=$ a$0$x$4+$ a$1$x$3+$ a$2$x$2+$ a$3$x $+$ a$4$ has at least one zero between

$0$ and $1.($Hint: Consider

f $($x$)=$ a$0$

$5$ x$5+$ a$1$

$4$ x$4+$ a$2$

$3$ x$3+$ a$3$

$2$ x$2+$ a$4$x$.$

Show that Rolle$$s theorem applies to f $($x$)$ on the interval $[0,1].$ Deduce that P $($x$)$ has a $0$ in

$[0,1]).$

$1$