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For $$15n^3+9n \in O\left(n^3\right)$$

I used the proof method to divide $f(n)$ by $g(n)$ to drive a constant of $24$ for all $n\:\ge \:1$.

However, I just realized the problem was asking to do it the direct way. I spent hours looking up the direct way for doing the big - o to find the positive real constant and non-negative integer, but can't find anything that isn't code.

Can someone help?

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    By $\xi$ do you mean $\in$? Also, the direct way of doing this type of problem is exactly what you did: finding a constant such as $24$ for $n \geq 1$. The indirect way would be to use some identities--like $n^a = O(n^b)$ for $a \leq b$--or the limit definition. – 2017-01-26
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    Okay, thank you for that verification. I needed it. – 2017-01-26

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