Final answer:
To solve the polynomial division using synthetic division, set up the synthetic division table and follow the steps to perform the division. The result of the division is x²+14x+48.
Step-by-step explanation:
To solve the polynomial division using synthetic division, we can use the coefficients of the given polynomial and the constant divisor. In this case, the polynomial is x³+12x²+20x-96 and the divisor is x-2.
First, set up the synthetic division table by writing the coefficients of the polynomial in descending order:
x³+12x²+20x-96
2 | 1 12 20 -96
Next, bring down the first coefficient, which is 1. Then multiply the divisor, 2, by the number in the quotient row and write the result under the next coefficient, which is 12. Add these two numbers to get the new value for the quotient row. Repeat this step for the remaining coefficients:
x³+12x²+20x-96
2 | 1 12 20 -96
2 28 96
----------------
1 14 48
The last number in the quotient row, 48, represents the constant term. Therefore, the result of the division is x²+14x+48.