Let's divide the polynomial [tex]\( x^2 + 5x - 2 \)[/tex] by [tex]\( x + 2 \)[/tex] using synthetic division step-by-step.
1. Synthetic Division Setup:
- Write down the coefficients of the polynomial [tex]\( x^2 + 5x - 2 \)[/tex]: [1, 5, -2]
- Identify the root of the divisor [tex]\( x + 2 \)[/tex]. Since [tex]\( x + 2 = 0 \)[/tex] implies [tex]\( x = -2 \)[/tex], we use [tex]\(-2\)[/tex] as the root for synthetic division.
2. Synthetic Division Process:
- Set up the synthetic division table:
```
-2 | 1 5 -2
```
- Bring down the leading coefficient (1) directly below the line.
```
-2 | 1 5 -2
----
1
```
- Multiply this number (1) by the root ([tex]\(-2\)[/tex]) and write the result under the next coefficient (5):
[tex]\(-2 \times 1 = -2\)[/tex]
```
-2 | 1 5 -2
----
1 -2
```
- Add the number above (5) to this product (-2) and write the sum below the line:
[tex]\(5 + (-2) = 3\)[/tex]
```
-2 | 1 5 -2
----
1 3
```
- Repeat the multiplication and addition steps for the remaining terms:
[tex]\(-2 \times 3 = -6\)[/tex]
```
-2 | 1 5 -2
----
1 3 -6
```
[tex]\(-2 + (-6) = -8\)[/tex]
3. Result Interpretation:
- The numbers below the line ([tex]\(1\)[/tex] and [tex]\(3\)[/tex]) represent the coefficients of the quotient polynomial.
- The last number ([tex]\(-8\)[/tex]) is the remainder.
[tex]\[
\frac{x^2 + 5x - 2}{x + 2} = x + 3 + \frac{-8}{x + 2}
\][/tex]
4. Formatting the Quotient and Remainder:
- Insert the remainder term correctly:
[tex]\[
\frac{x^2 + 5x - 2}{x + 2} = x + 3 - \frac{8}{x + 2}
\][/tex]
Thus, the correct multiple choice answer is:
D. [tex]\( x + 3 - \frac{8}{x + 2} \)[/tex]
So, the answer is D.
