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 evaluate \( f\left(\frac{1}{2}\right) \) using synthetic division and the Remainder Theorem, follow these steps:

1. **Set up synthetic division.**
   - First, rewrite the polynomial \( f(x) = 2x^5 + 5x^4 - 4x^2 + 3x - 1 \) and list the coefficients.
   - Since some terms are missing (e.g., \( x^3 \)), include zeros as placeholders.

   The polynomial \( f(x) \) has coefficients: \( [2, 5, 0, -4, 3, -1] \).

2. **Write down the synthetic division setup.**
   - We are dividing by \( x - \frac{1}{2} \), so use \( \frac{1}{2} \) as the synthetic divisor.

3. **Perform synthetic division.**
   - Bring down the first coefficient, \( 2 \).
   - Multiply \( 2 \) by \( \frac{1}{2} \) and add to the next coefficient.
   - Repeat this process across all coefficients.

Let's go through each step in the synthetic division process:

| Coefficients |      2       |      5      |      0       |      -4      |      3       |      -1      |
|--------------|--------------|-------------|--------------|--------------|--------------|--------------|
| Multiply     |              | \( 1 \)     | \( 3 \)      | \( 1.5 \)    | \( -1.25 \)  | \( 0.375 \)  |
| Add          | \( 2 \)      | \( 6 \)     | \( 3 \)      | \( -2.5 \)   | \( 1.75 \)   | **\( -0.625 \)** |

4. **Interpret the result.**
   - The remainder (last entry) after synthetic division is \( -0.625 \), which is the value of \( f\left(\frac{1}{2}\right) \) by the Remainder Theorem.

So, \( f\left(\frac{1}{2}\right) = -0.625 \).

Would you like further details on any part of this process? 

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### 5 Questions to Explore Further:
1. How is the Remainder Theorem generally used in evaluating polynomial functions?
2. Why are placeholders (like the \(0\) for \( x^3 \)) necessary in synthetic division?
3. What happens if the divisor is a different form, like \( x + a \), in synthetic division?
4. Can synthetic division be applied to non-polynomial expressions?
5. How does synthetic division differ from traditional polynomial long division?

---

**Tip:** Always check for missing terms in a polynomial and use zeros as placeholders in synthetic division to maintain accuracy.

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.

Discover how to find f(1/2) for the polynomial f(x) = 2x^5 + 5x^4 − 4x^2 + 3x − 1 using synthetic division and the Remainder Theorem. This step-by-step solution explores polynomial division techniques, the use of placeholders, and the calculation of remainders. Ideal for students in Grades 10-12.

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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