I recently came over a couple of questions while doing questions related to polynomial equations.
One of them struck me as quite peculiar:

If four distinct points on the curve $y=2x^4+7x^3+3x−5$ are collinear, then find the arithmetic mean of $X$-coordinates of the aforesaid points.

The solution of this involves a very basic application of the sum of roots of a polynomial equation. But, it was their solution that confused me. For applying the sum of roots property, the $X$-coordinates of the line that cut the curve must be the roots of the equation. If so, doesn't this line then become the $X$-axis itself? If so, how to prove that the $X$-axis is the line that cuts the curve? Also, for general cases of this question where the polynomial has a degree greater than or equal to two and the coefficients are any arbitrary real numbers, does the fact hold that any line cutting this curve at $n$ distinct points where $n$ is the degree of the equation will always be the $X$-axis?
{ I believe that this question came in the Putnam but the explanation over there confused me even more}