Question

Use synthetic division to find the result when 2x^4-15^3+27x^2-13x+15


is divided by x-5


A
2x3−5x2+2x−32x^3-5x^2+2x-32x
3
−5x
2
+2x−3

B
2x3+5x2−3x+22x^3+5x^2-3x+22x
3
+5x
2
−3x+2

C
10x3−2x2+5x−610x^3-2x^2+5x-610x
3
−2x
2
+5x−6

D
None of the above.

Answer 1

To perform synthetic division of the polynomial \(2x^4 - 15x^3 + 27x^2 - 13x + 15\) by \(x - 5\), we need to use the root \(5\) for synthetic division.

1. **Set up the synthetic division**:
- We will use the coefficients of the polynomial: \(2, -15, 27, -13, 15\).
- Write the root \(5\) on the left and the coefficients on the right.

```
5 | 2 -15 27 -13 15
```

2. **Perform the synthetic division**:
- Bring down the leading coefficient \(2\).

```
5 | 2 -15 27 -13 15
|
|________________
2
```

- Multiply \(5\) by \(2\) and write it under the next coefficient:
```
5 | 2 -15 27 -13 15
| 10
|________________
2 -5
```

- Add the column:
```
5 | 2 -15 27 -13 15
| 10
|________________
2 -5 27
```

- Multiply \(5\) by \(-5\) and write it under the next coefficient:
```
5 | 2 -15 27 -13 15
| 10 -25
|________________
2 -5 2
```

- Add the column:
```
5 | 2 -15 27 -13 15
| 10 -25
|________________
2 -5 2 -1
```

- Multiply \(5\) by \(2\) and write it under the next coefficient:
```
5 | 2 -15 27 -13 15
| 10 -25 10
|____________________
2 -5 2 -1
```

- Add the column:
```
5 | 2 -15 27 -13 15
| 10 -25 10
|____________________
2 -5 2 -1 5
```

3. **Result Interpretation**:
- The entries in the bottom row represent the coefficients of the quotient polynomial and the remainder.
- Thus, the result of dividing \(2x^4 - 15x^3 + 27x^2 - 13x + 15\) by \(x - 5\) is:
\[
2x^3 - 5x^2 + 2x - 1 \quad \text{with a remainder of } 0.
\]
- Therefore, we have:
\[
2x^3 - 5x^2 + 2x + 5.
\]

Now, let's check the given options:
- A: \(2x^3 - 5x^2 + 2x - 3\)
- B: \(2x^3 + 5x^2 - 3x + 2\)
- C: \(10x^3 - 2x^2 + 5x - 6\)
- D: None of the above.

The correct result \(2x^3 - 5x^2 + 2x + 5\) is not listed in options A, B, or C, so the correct answer is:

**D: None of the above.**