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The problem requires finding the value of \( f\left(-\frac{1}{2}\right) \) for the function 

\[
f(x) = 2x^4 - 9x^3 - 3x^2 + 3x + 8
\]

using synthetic division and the Remainder Theorem.

### Steps using Synthetic Division:

1. **Set up for synthetic division:** We divide the polynomial by \( x - \left(-\frac{1}{2}\right) \), so we use \( -\frac{1}{2} \) as the divisor in synthetic division.

2. **Coefficients of the polynomial:** Extract the coefficients of \( f(x) \), which are:
   \[
   [2, -9, -3, 3, 8]
   \]

3. **Perform synthetic division:**

   - Write down \( -\frac{1}{2} \) on the left.
   - Set up the row with the coefficients \( 2, -9, -3, 3, 8 \).
   - Bring down the first coefficient \( 2 \).
   - Multiply by \( -\frac{1}{2} \), and then add to the next coefficient:
     - \( 2 \times \left(-\frac{1}{2}\right) = -1 \); add to \( -9 \) to get \( -10 \).
     - \( -10 \times \left(-\frac{1}{2}\right) = 5 \); add to \( -3 \) to get \( 2 \).
     - \( 2 \times \left(-\frac{1}{2}\right) = -1 \); add to \( 3 \) to get \( 2 \).
     - \( 2 \times \left(-\frac{1}{2}\right) = -1 \); add to \( 8 \) to get \( 7 \).
     
4. **Interpret the result:** The remainder is \( 7 \), which is the value of \( f\left(-\frac{1}{2}\right) \) by the Remainder Theorem.

### Final Answer:
\[
f\left(-\frac{1}{2}\right) = 7
\]

Let me know if you need further details or explanations! Here are some follow-up questions for practice:

1. What is the Remainder Theorem, and how is it used in synthetic division?
2. How can synthetic division be applied to a divisor of the form \( x - c \)?
3. Can synthetic division be used for divisors of higher degree, like \( x^2 - 2x + 1 \)?
4. What happens if the remainder in synthetic division is zero?
5. How would the process change if you had to find \( f\left(\frac{1}{2}\right) \)?

### Tip:
The Remainder Theorem is useful because it allows you to quickly evaluate polynomials at specific values without directly plugging in the numbers!

Use synthetic division and the Remainder Theorem to find the indicated function value.

f(x) = 2x^4 - 9x^3 - 3x^2 + 3x + 8, f(-1/2) = ?

Learn how to find the value of f(-1/2) for the polynomial f(x) = 2x^4 - 9x^3 - 3x^2 + 3x + 8 using synthetic division and the Remainder Theorem. This step-by-step guide covers essential algebra concepts, making it ideal for high school students.

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