To find 
𝑓
(
1
/
2
)
f(1/2) using synthetic division and the Remainder Theorem, we will first perform synthetic division of the polynomial 
𝑓
(
𝑥
)
=
2
𝑥
5
+
5
𝑥
4
−
4
𝑥
2
+
3
𝑥
−
1
f(x)=2x 
5
 +5x 
4
 −4x 
2
 +3x−1 by 
𝑥
−
1
2
x− 
2
1
​
 .

Step 1: Set up synthetic division.

We will use 
1
2
2
1
​
  and the coefficients of 
𝑓
(
𝑥
)
f(x):

To use synthetic division to evaluate \( f\left(\frac{1}{2}\right) \) with the Remainder Theorem, we’ll set up synthetic division as follows:

### Step 1: Set up for synthetic division
1. **Identify the polynomial**: The polynomial is 
   \[
   f(x) = 2x^5 + 5x^4 - 4x^2 + 3x - 1
   \]
2. **Fill in missing terms**: Since there is no \( x^3 \) term, we’ll use a coefficient of \( 0 \) for that term.
   - Coefficients are: \( 2, 5, 0, -4, 3, -1 \)
3. **Set up the synthetic division**: We'll divide by \( x - \frac{1}{2} \), meaning we use \( \frac{1}{2} \) in synthetic division.

### Step 2: Perform synthetic division

Set up synthetic division as follows:

- Write down the coefficients: \( 2, 5, 0, -4, 3, -1 \)
- Use \( \frac{1}{2} \) to divide.

The process goes as follows:
1. Bring down the first coefficient, \( 2 \).
2. Multiply \( 2 \) by \( \frac{1}{2} \) to get \( 1 \), then add to the next coefficient \( 5 \), resulting in \( 6 \).
3. Multiply \( 6 \) by \( \frac{1}{2} \) to get \( 3 \), then add to \( 0 \), resulting in \( 3 \).
4. Multiply \( 3 \) by \( \frac{1}{2} \) to get \( 1.5 \), then add to \( -4 \), resulting in \( -2.5 \).
5. Multiply \( -2.5 \) by \( \frac{1}{2} \) to get \( -1.25 \), then add to \( 3 \), resulting in \( 1.75 \).
6. Multiply \( 1.75 \) by \( \frac{1}{2} \) to get \( 0.875 \), then add to \( -1 \), resulting in \( -0.125 \).

The last value, \( -0.125 \), is the remainder. According to the Remainder Theorem, this is also the value of \( f\left(\frac{1}{2}\right) \).

### Answer
\[
f\left(\frac{1}{2}\right) = -0.125
\]

Would you like more detailed steps or additional explanation?

---

1. What is synthetic division, and how is it used with polynomials?
2. How does the Remainder Theorem simplify evaluating polynomials?
3. Why is a missing term represented with a coefficient of zero in synthetic division?
4. Can synthetic division be used for divisors other than linear factors?
5. What are other methods to find \( f\left(\frac{1}{2}\right) \) besides synthetic division?

**Tip**: Remember, the Remainder Theorem provides a quick way to evaluate polynomials without fully expanding powers or substituting directly.

To find f(1/2) using synthetic division and the Remainder Theorem, we will first perform synthetic division of the polynomial f(x) = 2x^5 + 5x^4 − 4x^2 + 3x − 1 by x − 1/2.

Find f(1/2) for the polynomial f(x) = 2x^5 + 5x^4 - 4x^2 + 3x - 1 by using synthetic division and the Remainder Theorem. This solution provides a step-by-step guide to set up synthetic division, perform calculations, and apply the Remainder Theorem, covering essential polynomial concepts for high school math students.

Math Problem Statement

To find 𝑓 ( 1 / 2 ) f(1/2) using synthetic division and the Remainder Theorem, we will first perform synthetic division of the polynomial 𝑓 ( 𝑥 )

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