Answer:
2x^3 + 14x^2 - 13, with a remainder of -13.
Explanation:
First, let's write down the coefficients of the polynomial in descending order of the exponents:
2x^4 - 8x^3 - 27x^2 + 14x + 24
The coefficients are: 2, -8, -27, 14, 24.
Now, set up the synthetic division table:
6 | 2 -8 -27 14 24
To perform synthetic division, follow these steps:
Bring down the first coefficient, which is 2, into the leftmost box below the horizontal line:
6 | 2 -8 -27 14 24
|
2
Multiply the divisor, which is 6, by the number in the bottom box (2) and write the result below the next coefficient:
6 | 2 -8 -27 14 24
| 12
2
Add the number in the bottom box (2) to the number in the new box (12) and write the sum below the next coefficient:
6 | 2 -8 -27 14 24
| 12
2 14
Repeat steps 2 and 3 for the remaining coefficients:
6 | 2 -8 -27 14 24
| 12 48
2 14 -13
The number in the rightmost box is the remainder. In this case, the remainder is -13.
Now, let's write the result of the synthetic division:
2x^4 - 8x^3 - 27x^2 + 14x + 24 = (x - 6)(2x^3 + 14x^2 - 13)
Therefore, the result of dividing 2x^4 - 8x^3 - 27x^2 + 14x + 24 by x - 6 is 2x^3 + 14x^2 - 13, with a remainder of -13.