Final answer:
In synthetic division to remove the factor x+2 from the given polynomial x^3+7x^2+4x-12, we find the zero of the function (-2), perform synthetic division, and get a reduced polynomial of x^2 + 5x - 6.
Step-by-step explanation:
To remove the indicated factor x+2 from the polynomial x^3+7x^2+4x-12 using synthetic division, we first need to change the factor to the equivalent zero of the function, which is -2 in this case (because x would be -2 for the factor x+2 to equal zero).
Next, we set up the synthetic division:
- Write down the coefficients of the polynomial: 1, 7, 4, -12
- Place the zero (-2) to the left side:
-2 | 1 7 4 -12
| -2 -10 12
-----------------
1 5 -6 0
We multiply each number below the line by -2 and add to the number above it. The number at the bottom right corner is the remainder. A zero remainder means the factor was successfully removed. The other numbers give us the coefficients of the reduced polynomial: 1x^2 + 5x - 6. So after removing the factor x+2, we have the reduced polynomial x^2 + 5x - 6.