To find $f(33)$ for the polynomial $f(x) = 2x^5 - 8x^3 - 13x^2 - 25$ using the Remainder Theorem, we proceed as follows:

1. Apply the Remainder Theorem: According to the Remainder Theorem, $f(k)$ is equal to the remainder when $f(x)$ is divided by $x - k$.

   In this case, we need to calculate $f(33)$, which means we will substitute $x = 33$ directly into $f(x)$.

2. Substitute $x = 33$ in $f(x)$: $f(33) = 2(33)^5 - 8(33)^3 - 13(33)^2 - 25$

3. Calculate each term individually:

   • $33^5 = 39135393$
   • $33^3 = 35937$
   • $33^2 = 1089$

   Now substitute these values into the expression:

   $f(33) = 2(39135393) - 8(35937) - 13(1089) - 25$

4. Simplify each part:

   • $2 \times 39135393 = 78270786$
   • $8 \times 35937 = 287496$
   • $13 \times 1089 = 14157$

5. Combine all terms: $f(33) = 78270786 - 287496 - 14157 - 25$

6. Final Calculation: $f(33) = 77949108$

Therefore, $f(33) = 77949108$.

Would you like more details on this process? Here are some related questions:

1. How does the Remainder Theorem help with polynomial evaluations?
2. What steps would be needed if we used synthetic division?
3. How does the result change with different values of $k$?
4. Can we estimate large polynomial expressions without expanding fully?
5. Why are terms with zero coefficients omitted in polynomials?

Tip: For higher powers of polynomials, breaking down the calculations into smaller steps can reduce errors.