The correct setup box for the synthetic division is:
-1 | 1 -2 4
| 1 (coefficient of x^3) (coefficient of x^2) (coefficient of x) (constant term)
-(x +1)
| 1 (-1 * 1) (-1 * (-1) + 1) (-1 * 4 + 1)
------
| 1 -3 3 -3
Step 1: Set up the divisor and dividend coefficients.
| Divisor: x + 1 |
|---|---|
| Constant term: 1 |
| Dividend: x³ – 2x² + 4 |
|---|---|
| x³ coefficient: 1 |
| x² coefficient: -2 |
| x coefficient: 4 |
| Constant term: 4 |
Step 2: Place the divisor coefficient (negative coefficient of x) as the first term inside the table.
| -1: | 1 | -2 | 4 |
Step 3: Bring down the x³ coefficient of the dividend.
| -1: | 1 | -2 | 4 |
| ↓ | 1 | | |
Step 4: Multiply the top term by the brought-down coefficient and place the product below the next coefficient of the dividend (x²).
| -1: | 1 | -2 | 4 |
| ↓ | 1 | -1 | |
Step 5: Add the product and the next coefficient of the dividend. Write the sum below.
| -1: | 1 | -3 | 4 |
| ↓ | 1 | | |
Step 6: Repeat steps 4 and 5 for the remaining coefficients of the dividend.
| -1: | 1 | -3 | 3 |
| ↓ | 1 | | -3 |
Step 7: The bottom row represents the coefficients of the quotient polynomial in reverse order.
| -1: | 1 | -3 | 3 |
| ↓ | | 1 | -3 | -3 |
Therefore, the correct setup box for the synthetic division is:
-1 | 1 -2 4
| 1 (coefficient of x^3) (coefficient of x^2) (coefficient of x) (constant term)
-(x +1)
| 1 (-1 * 1) (-1 * (-1) + 1) (-1 * 4 + 1)
------
| 1 -3 3 -3