To determine if [tex]\(\frac{1}{2}\)[/tex] is a root of the polynomial [tex]\(P(x)=4x^4 - 8x^3 - 3x^2 + 7x - 2\)[/tex], we'll use synthetic division to divide the polynomial by [tex]\(x - \frac{1}{2}\)[/tex]. If [tex]\(\frac{1}{2}\)[/tex] is a root, the result should have a remainder of [tex]\(0\)[/tex].
Here are the steps for synthetic division of [tex]\(P(x)\)[/tex] by [tex]\(x - \frac{1}{2}\)[/tex]:
1. Write down the coefficients of the polynomial: [tex]\(4, -8, -3, 7, -2\)[/tex].
2. Set up the synthetic division. The divisor root is [tex]\(\frac{1}{2}\)[/tex], so we place [tex]\(\frac{1}{2}\)[/tex] to the left.
```
1/2 | 4 -8 -3 7 -2
| 2 -3 -3 2
-------------------
4 -6 -6 4 0
```
3. The process starts by bringing down the leading coefficient (4) to the bottom row.
4. Multiply [tex]\(\frac{1}{2}\)[/tex] by the value just written below the line (initially 4).
5. Write the result of the multiplication under the next coefficient ([tex]\(-8\)[/tex]) and add it to that coefficient.
6. Repeat the process for each coefficient.
Step-by-step calculation:
- Bring down 4.
```
4
```
- Multiply [tex]\(4\)[/tex] by [tex]\(\frac{1}{2}\)[/tex] to get 2. Write it under the next coefficient (-8) and add:
[tex]\(-8 + 2 = -6\)[/tex]
- Multiply [tex]\(-6\)[/tex] by [tex]\(\frac{1}{2}\)[/tex] to get [tex]\(-3\)[/tex]. Write it under the next coefficient (-3) and add:
[tex]\(-3 + (-3) = -6\)[/tex]
- Multiply [tex]\(-6\)[/tex] by [tex]\(\frac{1}{2}\)[/tex] to get [tex]\(-3\)[/tex]. Write it under the next coefficient (7) and add:
[tex]\(7 + (-3) = 4\)[/tex]
- Multiply [tex]\(4\)[/tex] by [tex]\(\frac{1}{2}\)[/tex] to get 2. Write it under the last coefficient (-2) and add:
[tex]\(-2 + 2 = 0\)[/tex]
The synthetic division shows that the remainder is [tex]\(0\)[/tex], which means that [tex]\(\frac{1}{2}\)[/tex] is a root of the polynomial.
Hence, [tex]\(\boxed{\text{root}}\)[/tex] of the polynomial.
