Divide using long division. (x3+6x2−x−30)÷(x−2) (1 point) Responses x2+8x−15 x squared plus 8 x minus 15 x2+8x+15 x squared plus 8 x plus 15 x2+8x+12, R 3 x2+8x+12, R 3 x2−8x+15
Let's use long division to divide the polynomial \(x^3 + 6x^2 - x - 30\) by \(x - 2\):
```
____________________
x - 2 | x^3 + 6x^2 - x - 30
- (x^3 - 2x^2)
______________
8x^2 - x
- (8x^2 -16x)
______________
15x - 30
- (15x - 30)
______________
0
```
Here is the step-by-step breakdown:
1. Divide \(x^3\) by \(x\) to get \(x^2\), and write it above the dividend.
2. Multiply \(x^2\) by the divisor \((x - 2)\) to get \(x^3 - 2x^2\), and subtract it from the dividend.
3. Bring down the next term of the dividend, which is \(-x\).
4. Divide the new term \(8x^2\) by \(x\) to get \(+8x\), and write it above the dividend.
5. Multiply \(8x\) by the divisor \((x - 2)\) to get \(8x^2 - 16x\), and subtract it from the remaining part of the dividend.
6. Bring down the next term of the dividend, which is \(-30\).
7. Divide \(15x\) by \(x\) to get \(+15\), and write it above the dividend.
8. Multiply \(15\) by the divisor \((x - 2)\) to get \(15x - 30\), and subtract it from the remaining part of the dividend.
After doing all these steps, the remainder is 0, and the quotient is \(x^2 + 8x + 15\).
So the result of the division is:
\(x^2 + 8x + 15\)