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 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.