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The problem asks us to use synthetic division and the Remainder Theorem to find \( f(2) \), where the polynomial function is given as:

\[
f(x) = 3x^3 - 10x^2 + 3x - 3
\]

### Steps using Synthetic Division:

1. **Set up the synthetic division table:**

   We divide the polynomial by \( x - 2 \), so we use 2 as the divisor. Write the coefficients of the polynomial \( 3x^3 - 10x^2 + 3x - 3 \), which are \( 3, -10, 3, -3 \).

   |   2   |  3  |  -10  |   3   |  -3   |
   |-------|-----|-------|-------|-------|
   |       |     |       |       |       |
   |-------|-----|-------|-------|-------|

2. **Perform the synthetic division:**
   - Bring down the first coefficient (3).
   - Multiply 3 by 2 and add to the next coefficient.
   - Continue the process.

   - Step 1: Bring down 3.
     \[
     3
     \]
   - Step 2: \( 3 \times 2 = 6 \), add to -10: \( -10 + 6 = -4 \).
     \[
     3 \quad -4
     \]
   - Step 3: \( -4 \times 2 = -8 \), add to 3: \( 3 + (-8) = -5 \).
     \[
     3 \quad -4 \quad -5
     \]
   - Step 4: \( -5 \times 2 = -10 \), add to -3: \( -3 + (-10) = -13 \).
     \[
     3 \quad -4 \quad -5 \quad -13
     \]

3. **Interpret the result:**
   The remainder is \( -13 \), so by the Remainder Theorem, \( f(2) = -13 \).

Thus, \( f(2) = -13 \).

Would you like further details or have any questions?

### Follow-up Questions:
1. What is synthetic division used for in algebra?
2. Can you describe the Remainder Theorem and its significance?
3. How do you check the result of synthetic division using substitution?
4. What is the relationship between roots and the Remainder Theorem?
5. How can synthetic division help in factoring polynomials?

**Tip:** Synthetic division only works when dividing by linear factors of the form \( x - c \).

Use synthetic division and the Remainder Theorem to find f(2) for f(x) = 3x^3 - 10x^2 + 3x - 3.

Learn how to use synthetic division and the Remainder Theorem to find f(2) for the polynomial f(x) = 3x^3 - 10x^2 + 3x - 3. This detailed explanation covers essential algebraic concepts and is suitable for Grades 10-12.

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