Points of Intersection

Question

Find (algebraically) the coordinates of all points of intersection of the graphs

f(x) = x⁴ + x² + 24x and 6x³ + 20

Answer 1

x(x^3 + x + 24)= 2(x^3 + 10)

x^4 -6x^3 +x^2 +24x -20 = 0

x=1 (1,0) is a point of intersection

divide the 4th degree polynomial by x-1 to get

x^3 - 5x^2 -4x +20 = 0

x=2 is a point of intersection (2,0)

divide x-2 into the cubic to get

x^2 -3x-10 =0

(x-5)(x+2) =0

x=5 x=-2

(5,0) (-2,0) are points of intersection