Question

use synthetic division to find the quotient. $\frac{x^{4}+2x^{2}-1}{x + 1}$ $\boldsymbol{?}x^{3}+\boldsymbol{?}x^{2}+\boldsymbol{?}x+\boldsymbol{?}+\frac{\boldsymbol{?}}{x + 1}$

Explanation:

Step1: Identify divisor root

For $x+1=0$, root is $x=-1$.
Coefficients of dividend: $1$ (for $x^4$), $0$ (for $x^3$), $2$ (for $x^2$), $0$ (for $x$), $-1$ (constant).

Step2: Set up synthetic division

$$\begin{array}{r|rrrrr} -1 & 1 & 0 & 2 & 0 & -1 \\ \hline & & -1 & 1 & -3 & 3 \\ \hline & 1 & -1 & 3 & -3 & 2 \end{array}$$

Step3: Extract quotient terms

The first four final coefficients are the coefficients of the quotient polynomial, and the last is the remainder.
Quotient polynomial: $1x^3 -1x^2 +3x -3$, remainder: $2$.

Answer:

$1x^3 + (-1)x^2 + 3x + (-3) + \frac{2}{x+1}$
Or simplified: $x^3 - x^2 + 3x - 3 + \frac{2}{x+1}$