Question

use synthetic division to divide the two polynomials.
$z + 1 \overline{) -5z^3 - z^2 + 4z}$
is the divisor given in $(x - r)$ form? yes
how many terms are in the dividend? 4
enter the quotient and remainder.
quotient + remainder
synthetic division table
-1
-5 -1 4 0
-5

Explanation:

Step1: Set up synthetic division

Divisor $z+1 = z - (-1)$, so $r=-1$. Coefficients of dividend $-5z^3 - z^2 + 4z + 0$ are $-5, -1, 4, 0$.

$$\begin{array}{r|rrrr} -1 & -5 & -1 & 4 & 0 \\ \hline & & & & \\ \end{array}$$

Step2: Bring down leading coefficient

Bring down $-5$ to the bottom row.

$$\begin{array}{r|rrrr} -1 & -5 & -1 & 4 & 0 \\ \hline & & & & \\ & -5 & & & \\ \end{array}$$

Step3: Multiply and add (first column)

Multiply $-5$ by $-1$, add to $-1$.
$\text{Calculation: } (-5)(-1) + (-1) = 5 - 1 = 4$

$$\begin{array}{r|rrrr} -1 & -5 & -1 & 4 & 0 \\ \hline & & 5 & & \\ & -5 & 4 & & \\ \end{array}$$

Step4: Multiply and add (second column)

Multiply $4$ by $-1$, add to $4$.
$\text{Calculation: } 4(-1) + 4 = -4 + 4 = 0$

$$\begin{array}{r|rrrr} -1 & -5 & -1 & 4 & 0 \\ \hline & & 5 & -4 & \\ & -5 & 4 & 0 & \\ \end{array}$$

Step5: Multiply and add (third column)

Multiply $0$ by $-1$, add to $0$.
$\text{Calculation: } 0(-1) + 0 = 0 + 0 = 0$

$$\begin{array}{r|rrrr} -1 & -5 & -1 & 4 & 0 \\ \hline & & 5 & -4 & 0 \\ & -5 & 4 & 0 & 0 \\ \end{array}$$

Step6: Identify quotient and remainder

Bottom row coefficients $-5, 4, 0$ form quotient $-5z^2 + 4z$, last value $0$ is remainder.

Answer:

Quotient: $\boldsymbol{-5z^2 + 4z}$
Remainder: $\boldsymbol{0}$

Filled synthetic division table:

$$\begin{array}{r|rrrr} -1 & -5 & -1 & 4 & 0 \\ \hline & & 5 & -4 & 0 \\ & -5 & 4 & 0 & 0 \\ \end{array}$$