Question

How many `x` intercepts appear on the graph of `f(x)=x^4-x^3+x^2-x`?

Answer 1

We have only two intercepts on `x`-axis - `0` and `1`.

Explanation:

`f(x)=x^4-x^3+x^2-x`

= `x^3(x-1)+x(x-1)`

= `(x^3+x)(x-1)`

= `x(x^2+1)(x-1)`

Hence, `f(x)=0` only when `x=0` and `x=1` and `(x^2+1)` is always positive and never less than `1`.

Hence we have only two intercepts on `x`-axis - `0` and `1`.
graph{x^4-x^3+x^2-x [-9.75, 10.25, -1.6, 8.4]}