Question

At which root does the graph of `f(x) = (x-5)^3(x+2)^2` touch the `x` axis?

Answer 1

`f(x)` touches (but does not cross) the X-axis at `x=-2`

Explanation:

Since `(x-5)` is raised to the third power, negative values for `(x-5)` will result in negative values for `f(x)` and positive values for `(x-5)` will result in positive values for `f(x)`

Since the `(x+2)` factor is squared it's value will never be less than zero and therefore it will not cause `f(x)` to cross the X-axis.