I've stumbled upon some questions while practicing polynomials. For each question, I can find the solution, but I think there should be an easier way. For example: x7 + x + 1 = 0. I'm able to find the (real) solutions of this equation by using methods like Horner's. However, is there an easier way to find the number of real solutions this equation has. I don't need to know the real solutions, just the number of them. Next: x4 - 4x3 + 2x2 + 2x - 1 = 0. I have given that their are 4 real solutions. The question: How many of those solutions are smaller than 0. Ofcourse I can compute them all, but is their a faster, more intuitive way for finding the answer to this question? Lastly, how can I determine how many polynomial functions exist through some given points. For example, how many (polynomial) functions do their exist through the points: (0, 0), (1, 1), (2, 4) and (3, 9). My intuition says one solution (y = x2), but is their a way to calculate this using arithmetic? Are their overal rules/exercises to improve intuition for this kind of functions and their roots/real solutions? Thank you in advance! PS: I'm not a native English speaker, so some translations/terminology may be wrong, don't hesitate to correct and/or ask what I mean if it's not clear.