Question

Use synthetic division to find the result when 2x^4-15x^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 use synthetic division to divide \( 2x^4 - 15x^3 + 27x^2 - 13x + 15 \) by \( x - 5 \), we need to set up the synthetic division with the root of \( x - 5 \), which is \( 5 \).

Here's how we perform synthetic division step-by-step:

1. Write down the coefficients of the polynomial:
- \( 2, -15, 27, -13, 15 \)

2. Set up the synthetic division:
- We put \( 5 \) on the left, and the coefficients on the right.

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

3. Bring down the leading coefficient (which is \( 2 \)):
```
5 | 2 -15 27 -13 15
|_______
2
```

4. Multiply \( 5 \) (the root) by \( 2 \) and add to the next coefficient:
- \( 5 \times 2 = 10 \)
- \( -15 + 10 = -5 \)

```
5 | 2 -15 27 -13 15
| 10
|_______
2 -5
```

5. Repeat the process:
- Multiply \( 5 \) by \( -5 \):
- \( 5 \times -5 = -25 \)
- \( 27 - 25 = 2 \)

```
5 | 2 -15 27 -13 15
| 10 -25
|_______
2 -5 2
```

6. Continue:
- Multiply \( 5 \) by \( 2 \):
- \( 5 \times 2 = 10 \)
- \( -13 + 10 = -3 \)

```
5 | 2 -15 27 -13 15
| 10 -25 10
|_______
2 -5 2 -3
```

7. Final step:
- Multiply \( 5 \) by \( -3 \):
- \( 5 \times -3 = -15 \)
- \( 15 - 15 = 0 \)

```
5 | 2 -15 27 -13 15
| 10 -25 10 -15
|_______
2 -5 2 -3 0
```

The result from synthetic division is:
- The coefficients of the quotient \( 2x^3 - 5x^2 + 2x - 3 \) and a remainder of \( 0 \).

Thus, when \( 2x^4 - 15x^3 + 27x^2 - 13x + 15 \) is divided by \( x - 5 \), the result is:

\[
\boxed{A} \quad 2x^3 - 5x^2 + 2x - 3
\]