I am going to pick 1, 2, and 3 as roots. The cubic polynomial that does that is:
I am going to show that the line tangent to the cubic at passes through third root of the cubic, (3, 0).
The slope of the required line is:
evaluated at x = 3/2:
We know this line is to be tangent to the cubic at x = 3/2, so we need to find the correct y-intecept of this line:
So we need to know which y corresponds to the cubic with a value of x = 3/2:
Plugging this into the line equation:
So our tangent line is
Now, this presumably crosses the x-axis at the same point as the third root of the cubic: (3, 0). Let's find out:
So the line does indeed intersect the point (3, 0). The graph below gives a visual indication of this.