The table in the exercise shows a list of expressions that represent processing times $T(n)$ of algorithms for solving problems of size $n$. The task is to find the dominant term(s) for each expression and specify the Big-O complexity based on that.

Let's go through each expression and determine the dominant term and its corresponding Big-O notation:

1. Expression: $5 + 0.001n^3 + 0.025n$

   • Dominant term: $0.001n^3$
   • Big-O: $O(n^3)$

2. Expression: $500n + 100n^{1.5} + 50n \log_{10} n$

   • Dominant term: $100n^{1.5}$
   • Big-O: $O(n^{1.5})$

3. Expression: $0.3n + 5n^{1.5} + 2.5 \cdot n^{1.75}$

   • Dominant term: $2.5 \cdot n^{1.75}$
   • Big-O: $O(n^{1.75})$

4. Expression: $n^2 \log_2 n + n (\log_2 n)^2$

   • Dominant term: $n^2 \log_2 n$
   • Big-O: $O(n^2 \log n)$

5. Expression: $n \log_3 n + n \log_2 n$

   • Dominant term: $n \log_2 n$
   • Big-O: $O(n \log n)$

6. Expression: $3 \log_3 n + \log_2 \log_2 \log_2 n$

   • Dominant term: $3 \log_3 n$
   • Big-O: $O(\log n)$

7. Expression: $100n + 0.01n^2$

   • Dominant term: $0.01n^2$
   • Big-O: $O(n^2)$

8. Expression: $0.01n + 100n^2$

   • Dominant term: $100n^2$
   • Big-O: $O(n^2)$

9. Expression: $2n + n^{0.5} + 0.5n^{1.25}$

   • Dominant term: $0.5n^{1.25}$
   • Big-O: $O(n^{1.25})$

10. Expression: $0.01n \log_2 n + n(\log_2 n)^2$

    • Dominant term: $n (\log_2 n)^2$
    • Big-O: $O(n (\log n)^2)$

11. Expression: $100n \log_3 n + n^3 + 100n$

    • Dominant term: $n^3$
    • Big-O: $O(n^3)$

12. Expression: $0.003n \log_3 n + \log_2 \log_2 n$

    • Dominant term: $0.003n \log_3 n$
    • Big-O: $O(n \log n)$

Would you like more details or have any questions? Here are some follow-up questions to consider:

1. How is the dominant term used to determine Big-O complexity?
2. Why does the constant coefficient not matter in Big-O notation?
3. What is the difference between $O(n \log n)$ and $O(n (\log n)^2)$?
4. Can two algorithms with different Big-O complexities ever have similar performance for small inputs?
5. What is the significance of logarithmic terms like $\log_2 n$ in algorithm analysis?

Tip: When analyzing algorithms, focus on how terms behave as $n$ grows large, since Big-O notation is concerned with asymptotic behavior.