Question

use synthetic division to find the remainder.
$\frac{x^{4} + x^{3} - x^{2} + x - 3}{x + 2}$

Explanation:

Step1: Identify root of divisor

Set $x+2=0$, so $x=-2$.
List coefficients of dividend: $1, 1, -1, 1, -3$

Step2: Set up synthetic division

Bring down the leading coefficient:
$1$

Step3: Multiply and add

Multiply $1 \times (-2) = -2$, add to next coefficient:
$1 + (-2) = -1$

Step4: Multiply and add

Multiply $-1 \times (-2) = 2$, add to next coefficient:
$-1 + 2 = 1$

Step5: Multiply and add

Multiply $1 \times (-2) = -2$, add to next coefficient:
$1 + (-2) = -1$

Step6: Multiply and add (find remainder)

Multiply $-1 \times (-2) = 2$, add to last coefficient:
$-3 + 2 = -1$

Answer:

$-1$