Question

`F(x)=x^6-x-1` cuts the `x` axis at how many points?

Answer 1

`2`

Explanation:

Given:

`F(x) = x^6-x-1`

The pattern of signs of the coefficients of `F(x)` is `+ - -`. With one change of signs, Descartes' Rule of Signs tells us that `F(x)` has exactly one positive real zero.

The pattern of signs of the coefficients of `F(-x)` is `+ + -`. With one change of signs, Descartes' Rule of Signs tells us that `F(x)` has exactly one negative real zero.

So `F(x)` intersects the `x` axis at two points - one negative and one positive.

graph{x^6-x-1 [-2.2, 2.2, -3.24, 6.76]}