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!