Problem - 2856
Consider the lines that meet the graph $y = 2x^4 + 7x^3 + 3x - 5$ in four distinct points $P_i = (x_i, y_i), i = 1, 2, 3, 4$. Prove that
$$\frac{x_1 + x_2 + x_3 + x_44}{4}$$
is independent of the line, and compute its value.
Let the line be $y=mx + b$. Then the $x$-coordinates of the four intersection points must be the for zeros of $$(2x^4 + 7x^3 + 3x-5)-(mx+b)=2x^4+7x^3 + (3-m)x -(5+b)$$
It follows that, by the Vieta's theorem, $$\frac{x_1 + x_2+x_3+x_4}{4}=\frac{1}{4}\times\left(-\frac{7}{2}\right)=\boxed{-\frac{7}{8}}$$