Show that if four distinct points of the curve $y=2x^4+7x^3+3x-5$ are collinear then their average x-coordinate is some constant k. Find k.
Shall I use vector to calculate their x-coordinate, or shall I use coordinate geometry? I always find geometry of this type challenging, could someone give me some advice on how to improve my ability on that?
The calculation of the $x$-coordinates is a distraction posed by the question writer. Collinear points are determined by the intersection of the polynomial and some line $y=mx+b$. Subtracting the equations gives us a new polynomial whose roots are exactly the $x$-coordinates of interest. The average is just the sum divided by 4, and the sum is attainable with Vieta's formulas as $-7/2$ (the higher degree terms don't change in the subtraction). Hence the average is always $-7/8$.
If you're unfamiliar with Vieta's formulas, you can observe that every fourth degree polynomial has 4 roots, so it can be written as $$\alpha(x-a)(x-b)(x-c)(x-d) = \alpha x^4 - \alpha (a+b+c+d)x^3 + \ldots.$$ Dividing the coefficients of the two highest power terms gives you the negative of the sum of the roots.