To find the quotient in polynomial form, let's perform synthetic division step by step. The given polynomial to be divided is [tex]\( 1x^2 + 6x - 16 \)[/tex] by [tex]\( x - 2 \)[/tex]. The root of the divisor polynomial [tex]\( x - 2 \)[/tex] is [tex]\( 2 \)[/tex].
### Steps to Perform Synthetic Division:
1. Setup the Synthetic Division:
- Write down the coefficients of the polynomial: [tex]\( [1, 6, -16] \)[/tex].
- Write the root of the divisor to the left: [tex]\( 2 \)[/tex].
```
2 | 1 6 -16
----------------
```
2. Start the Synthetic Division Process:
- Bring down the leading coefficient (1) to the bottom row.
```
2 | 1 6 -16
----------------
1
```
3. Multiply and Add:
- Multiply the root (2) by the value just written below the line (1), then add this result to the next coefficient (6).
[tex]\[
2 \times 1 = 2
\][/tex]
[tex]\[
6 + 2 = 8
\][/tex]
- Write the result below the line under the second coefficient.
```
2 | 1 6 -16
----------------
1 8
```
4. Repeat the Multiply and Add Step:
- Multiply the root (2) by the new value (8), then add this to the next coefficient (-16).
[tex]\[
2 \times 8 = 16
\][/tex]
[tex]\[
-16 + 16 = 0
\][/tex]
- Write the result below the line under the last coefficient.
```
2 | 1 6 -16
----------------
1 8 0
```
5. Determine the Quotient and the Remainder:
- The bottom row (excluding the last value) represents the coefficients of the quotient polynomial. Here, the last value (0) is the remainder.
- So, the quotient polynomial has coefficients: [tex]\( [1, 8] \)[/tex], corresponding to [tex]\( x + 8 \)[/tex].
### Conclusion:
Thus, the quotient in polynomial form is [tex]\( x + 8 \)[/tex].
From the options given:
A. [tex]\( x - 6 \)[/tex]
B. [tex]\( x - 8 \)[/tex]
C. [tex]\( x + 8 \)[/tex]
D. [tex]\( x + 6 \)[/tex]
The correct answer is:
C. [tex]\( x + 8 \)[/tex]
