divide. if there is a remainder, include it as a simplified fraction.
$(z^4 + 9z^3 + 19z^2 - 5z) \div (z + 5)$

Question

Accepted Answer

# Explanation:
## Step1: Use polynomial long division
Divide the first term of the dividend $z^4$ by the first term of the divisor $z$ to get $z^3$. Multiply the divisor $(z + 5)$ by $z^3$ to get $z^4 + 5z^3$. Subtract this from the dividend:
$$
\begin{align*}
&(z^4 + 9z^3 + 19z^2 - 5z) - (z^4 + 5z^3)\
=&z^4 + 9z^3 + 19z^2 - 5z - z^4 - 5z^3\
=&4z^3 + 19z^2 - 5z
\end{align*}
$$
## Step2: Divide the new first term $4z^3$ by $z$ to get $4z^2$. Multiply the divisor by $4z^2$ to get $4z^3 + 20z^2$. Subtract:
$$
\begin{align*}
&(4z^3 + 19z^2 - 5z) - (4z^3 + 20z^2)\
=&4z^3 + 19z^2 - 5z - 4z^3 - 20z^2\
=& -z^2 - 5z
\end{align*}
$$
## Step3: Divide the new first term $-z^2$ by $z$ to get $-z$. Multiply the divisor by $-z$ to get $-z^2 - 5z$. Subtract:
$$
\begin{align*}
&(-z^2 - 5z) - (-z^2 - 5z)\
=&-z^2 - 5z + z^2 + 5z\
=&0
\end{align*}
$$
# Answer:
$z^3 + 4z^2 - z$

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QUESTION IMAGE

Question

Explanation:

Step1: Use polynomial long division

Step2: Divide the new first term \(4z^3\) by \(z\) to get \(4z^2\). Multiply the divisor by \(4z^2\) to get \(4z^3 + 20z^2\). Subtract:

Step3: Divide the new first term \(-z^2\) by \(z\) to get \(-z\). Multiply the divisor by \(-z\) to get \(-z^2 - 5z\). Subtract:

Answer: