jmc

Question

The graphs of y = x^3 - 3x + 2 and x + 4y = 4 intersect in the points (x_1,y_1), (x_2,y_2), and (x_3,y_3). If x_1 + x_2 + x_3 = A and y_1 + y_2 + y_3 = B, compute the ordered pair (A,B).

Accepted Answer

From x + 4y = 4, y = -x4 + 1. Hence, the x_i are the roots of x^3 - 3x + 2 = -x4 + 1.Then by Vieta's formulas, x_1 + x_2 + x_3 = 0, and y_1 + y_2 + y_3 = -x_14 + 1 - x_24 + 1 - x_34 + 1 = -x_1+x_2+x_34+3 = 3.Hence, (A,B) = (0,3).