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 )

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?

---

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.