# Emphasizing the Coefficients of the Leading Order and Using Big O Notation for the Remainder

I am trying to understand the correct application of Big O notation to polynomial expressions, including terms with negative coefficients. For example, consider the polynomial $$2n^3-2n^2+n+1$$, where $$n$$ is a positive number. When emphasizing the coefficient of the leading term and using Big O notation for the remainder, can this expression be written as: $$2n^3 - O(n^2)$$? I am unsure if this notation is permissible in scientific writing.

Additionally, I would like to know if functions like $$2n^3-2n^2+n+1$$ can be written as: $$2n^3+O(n)$$ while ignoring the negative terms.

Any insights or guidance on this matter would be greatly appreciated.

$$f(n) = 2n^3 - 2n^2 + n + 1$$ can be written as $$f(n) = 2n^3 - g(n)$$ where $$g(n) = 2n^2 - n - 1$$. Now $$g(n) \in O(n^2)$$ as well as $$g(n) \in \Omega(n^2)$$. Then you may loosely write $$2n^3 - \Omega(n^2) \le f(n) \le 2n^3 - O(n^2)$$. In my opinion it is better to write $$f(n) = 2n^3 - \Theta(n^2)$$.
$$f(n) = 2n^3 - 2n^2 + n + 1 \le 2n^3 + n$$. Thus $$f(n) \le 2n^3 + O(n)$$.