Question

Which synthetic division shows how to find the quotient?

x5−4x3+2x+6÷x+2


(1 point)

Answer 1

To perform synthetic division for the polynomial \( x^5 - 4x^3 + 2x + 6 \) by \( x + 2 \), we first rewrite \( x + 2 \) to find the root, which is \( -2 \).

1. **Set up the synthetic division**:
- The coefficients of the polynomial \( x^5 - 4x^3 + 2x + 6 \) are:
- \( 1 \) (for \( x^5 \))
- \( 0 \) (for \( x^4 \), since there's no \( x^4 \) term)
- \( -4 \) (for \( x^3 \))
- \( 0 \) (for \( x^2 \), since there's no \( x^2 \) term)
- \( 2 \) (for \( x^1 \))
- \( 6 \) (for the constant term)

The coefficients are \([1, 0, -4, 0, 2, 6]\).

2. **Write down the synthetic division setup**:
- The root \( -2 \) is placed on the left, and the coefficients are lined up on the right:

```
-2 | 1 0 -4 0 2 6
|
|_________________________
```

3. **Start the synthetic division process**:
- Bring down the leading coefficient \( 1 \):

```
-2 | 1 0 -4 0 2 6
|
|_________________________
1
```

- Now multiply \( 1 \) by \( -2 \) and add to the next coefficient \( 0 \):

```
-2 | 1 0 -4 0 2 6
| -2
|_________________________
1 -2
```

- Next, multiply \( -2 \) by \( -2 \) and add to the next coefficient \( -4 \):

```
-2 | 1 0 -4 0 2 6
| -2 4
|_________________________
1 -2 0
```

- Next, multiply \( 0 \) by \( -2 \) and add to the next coefficient \( 0 \):

```
-2 | 1 0 -4 0 2 6
| -2 4 0
|_________________________
1 -2 0 2
```

- Next, multiply \( 2 \) by \( -2 \) and add to the next coefficient \( 6 \):

```
-2 | 1 0 -4 0 2 6
| -2 4 0 -4
|_________________________
1 -2 0 2 2
```

4. **Finalize the synthetic division**:
- The resulting row represents the coefficients of the quotient polynomial and the remainder. The quotient polynomial is:
\[
x^4 - 2x^3 + 0x^2 + 2x + 2
\]
- The remainder is \( 2 \).

Thus, the final result of the synthetic division is:
\[
\text{Quotient: } x^4 - 2x^3 + 2x + 2 \quad \text{Remainder: } 2
\]

To summarize:
\[
\frac{x^5 - 4x^3 + 2x + 6}{x + 2} = x^4 - 2x^3 + 2x + 2 + \frac{2}{x + 2}
\]