This question occurred to me from an earlier question I had asked (Use of calculus in finding relations between roots of polynomial equations of degree greater than or equal to 2.)
Now , if the curve $p(x)=ax^4 + bx^3 + cx^2 + dx + e$ is cut by a line at four distinct points , then the $x$-coordinates of the intersection points are the roots of my original function $p(x)$. From this, is it true that for any polynomial $q(x)$ with degree $n$, when cut by another curve $f(x)$ of degree not greater than $n-2$, the $x$-coordinates of the intersection points are the roots of my original function $q(x)$? And if it is, then can I just apply the sum of roots property to calculate the sum of these $x$-coordinates?