Big O notation changing the power

Question

I want to Express each of the following functions using Big-O notation.

a(n) = 2n + 3n^2 + nlog(n)

b(n) = 5nlog(n) + 10n^3 + n^2

for a(n) I assumed that the answer would be O(n^2) However apparently it is O(n^3) this is the same for b(n) where I assumed the notation would be O(n^3) however it is O(n^4). Is it a rule to round up the power when writing the notation? Why would this be the case? Isn't the notation supposed to take the upper-bound?

Answer 1

You are right, a(n) = O(n²) and b(n) = O(n³).

However, notice that a(n) is also O(n³) and indeed O(n¹⁰⁰⁰). Usually though, we want to express the tightest bound we can find.