Use of calculus in finding relations between roots of polynomial equations of degree greater than or equal to $2$ 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}
 A: Suppose the four points are on the quartic $y=q(x)$ and the line $y=mx+c$ then they satisfy $$p(x)=q(x)-mx-c=0$$ a quartic equation which has the same coefficients of $x^4$ and $x^3$ as $q(x)$. So the sum of the roots of $p(x)$ (Vieta relations) is the same as for $q(x)$.

This applies to polynomial equations of any degree above $2$.
A: Take any $4$ points on the curve. They are collinear implies there is a line $y=ax+b$ through these $4$ points. Then the $x$-coordinates of these $4$ points will be the roots of the equation $2x^4+7x^3+3x-5-ax-b=0$. And sum of the roots of this equation will be $-7/2$, independent of $a,b$.
A: 
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?

No. Consider polynomial function $W(x)$ with degree $n \geq2$ that has n real roots.
W(x) + x is polynomial function of same degree and is cut by the line y = x in n points, roots of polynomial W.
