Question

Use synthetic division to divide 6x^3-10x^2+20 by x+1
.

A
6x2−4x−4+16x+16x^2-4x-4+\frac{16}{x+1}6x
2
−4x−4+
x+1
16
​


B
6x2−16x+166x^2-16x+166x
2
−16x+16 +1

C
6x2−16x+16−4x+16x^2-16x+16-\frac{4}{x+1}6x
2
−16x+16−
x+1
4
​


D
6x2−16x+16+4x+16x^2-16x+16+\frac{4}{x+1}6x
2
−16x+16+
x+1
4
​

Answer 1

To divide \( 6x^3 - 10x^2 + 20 \) by \( x + 1 \) using synthetic division, we first write the coefficients of the polynomial and the root of the divisor.

The coefficients of \( 6x^3 - 10x^2 + 0x + 20 \) are:
- \( 6 \) (for \( x^3 \))
- \( -10 \) (for \( x^2 \))
- \( 0 \) (for \( x \))
- \( 20 \) (constant)

The root corresponding to \( x + 1 = 0 \) is \( -1 \).

Now we set up synthetic division:

1. Write down the coefficients:
\[
6, -10, 0, 20
\]
2. Write \(-1\) to the left.

```
-1 | 6 -10 0 20
|
```

3. Bring down the leading coefficient (first number):

```
-1 | 6 -10 0 20
|
| 6
```

4. Multiply \(-1\) by the value just written below the line (6), and write the result under the next coefficient (-10):

```
-1 | 6 -10 0 20
| -6
| 6 -16
```

5. Add the column: \(-10 - 6 = -16\).

6. Repeat this process. Multiply \(-1\) by \(-16\) and write it under \(0\):

```
-1 | 6 -10 0 20
| -6 16
| 6 -16 16
```

7. Add the column: \(0 + 16 = 16\).

8. Repeat. Multiply \(-1\) by \(16\) and write under \(20\):

```
-1 | 6 -10 0 20
| -6 16 -16
| 6 -16 16 4
```

9. Finally, add: \(20 - 16 = 4\).

Now we have the result of the synthetic division:

- The resulting coefficients are \( 6, -16, 16 \).
- The remainder is \( 4 \).

Thus, we can express the result as:

\[
6x^2 - 16x + 16 + \frac{4}{x + 1}
\]

Looking at the options provided:

- **A:** \( 6x^2 - 4x - 4 + \frac{16}{x + 1} \)
- **B:** \( 6x^2 - 16x + 16 + 1 \)
- **C:** \( 6x^2 - 16x + 16 - \frac{4}{x + 1} \)
- **D:** \( 6x^2 - 16x + 16 + \frac{4}{x + 1} \)

The correct answer is **D**:

\[
\boxed{6x^2 - 16x + 16 + \frac{4}{x + 1}}
\]