Given 2 polynomials with degree m and n, what is the max number of real solutions when the 2 polynomials are crossing over?
- PuzzlingLv 75 days agoFavorite Answer
Let's just take an example with m = 3, n = 2.
Polynomial of degree 3:
ax³ + bx² + cx + d
Polynomial of degree 2:
ex² + fx + g
To find the points of intersection, we would equate them.
ax³ + bx² + cx + d = ex² + fx + g
Get everything on one side:
ax³ + bx² - ex² + cx - fx + d - g = 0
ax³ + (b - e)x² + (c - f)x + (d - g) = 0
That results in a new 3rd degree polynomial and you are looking for the number of zeros. That can't be more than 3.
The answer, therefore, is the maximum of the degrees m and n.
Caveat: There is one degenerate case to consider where the two polynomials are the same polynomial and there they would have infinitely many points where they coincide. But if you assume they have different degrees then they can't intersect more than the degree of the highest degree polynomial.