Not necessarily, there are maximum 3 but not necessarily 3 points. However, I see that I have indeed made an error, for which I'm very sorry. The correct answer should be:

The points of intersection can be found for those values x when:

1/2 x³ + x² + 1 = 9x² - 57/2 x +1

=> 1/2 x³ - 8x² + 57/2x = 0

=> x (x² - 16x + 57) = 0

=> x = 0 or x = 8+√7 or x = 8-√7

So there are indeed 3 points of intersection: (0,1) , (8+√7, 412 + 231/2 √7) and (8-√7, 412 - 231/2 √7) (or with rounded numbers (0,1), (10.646, 717.584) and (5.354, 106.416) )

Again my apologies for this error (no need to accept this answer off course!)