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 \( x^5 - 4x^3 + 2x + 6 \) divided by \( x + 2 \), we need to use the root of the divisor \( x + 2 \). The root is \( -2 \).

Here are the steps for synthetic division:

1. **Set Up**: Write down the coefficients of the polynomial. For \( x^5 - 4x^3 + 2x + 6 \), the coefficients are:
- For \( x^5 \): 1
- For \( x^4 \): 0 (since there is no \( x^4 \) term)
- For \( x^3 \): -4
- For \( x^2 \): 0 (since there is no \( x^2 \) term)
- For \( x^1 \): 2
- For the constant term: 6

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

2. **Synthetic Division Process**:

- Write down -2 (the value of the root).
- Bring down the leading coefficient (1).

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

- Multiply -2 by 1 (the number brought down) to get -2, and write it under the next coefficient.

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

- Now add down the column: \( 0 + (-2) = -2 \).

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

- Repeat this process: Multiply -2 by -2 to get 4, write it under the next coefficient, and add down.

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

- Next, multiply -2 by 0 to get 0 and add to the next coefficient.

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

- Finally, multiply -2 by 2 to get -4 and add to the last coefficient.

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

3. **Result**: The bottom row gives us the coefficients of the quotient polynomial and the remainder.

The result is:
- Quotient: \( 1x^4 - 2x^3 + 0x^2 + 2x + 0 \) or simplified: \( x^4 - 2x^3 + 2x \)
- Remainder: 0

So, the final answer to \( x^5 - 4x^3 + 2x + 6 \) divided by \( x + 2 \) is:

\[
\boxed{x^4 - 2x^3 + 2x}
\] (with a remainder of \( 0 \)).