Question

1. $(n^3 + n^2 + 8n + 20) div (n + 2)$

Explanation:

Step 1: Use polynomial long division

Divide the first term of the dividend \(n^3\) by the first term of the divisor \(n\) to get \(n^2\). Multiply the divisor \(n + 2\) by \(n^2\) to get \(n^3 + 2n^2\). Subtract this from the dividend:
\[

$$\begin{align*} &(n^3 + n^2 + 8n + 20) - (n^3 + 2n^2)\\ =&n^3 + n^2 + 8n + 20 - n^3 - 2n^2\\ =& -n^2 + 8n + 20 \end{align*}$$

\]

Step 2: Divide the new first term \(-n^2\) by \(n\)

We get \(-n\). Multiply the divisor \(n + 2\) by \(-n\) to get \(-n^2 - 2n\). Subtract this from the previous result:
\[

$$\begin{align*} &(-n^2 + 8n + 20) - (-n^2 - 2n)\\ =& -n^2 + 8n + 20 + n^2 + 2n\\ =& 10n + 20 \end{align*}$$

\]

Step 3: Divide the new first term \(10n\) by \(n\)

We get \(10\). Multiply the divisor \(n + 2\) by \(10\) to get \(10n + 20\). Subtract this from the previous result:
\[

$$\begin{align*} &(10n + 20) - (10n + 20)\\ =& 10n + 20 - 10n - 20\\ =& 0 \end{align*}$$

\]

Step 4: Combine the results

The quotient is \(n^2 - n + 10\) and the remainder is \(0\). So, \((n^3 + n^2 + 8n + 20)\div(n + 2)=n^2 - n + 10\)

Answer:

\(n^2 - n + 10\)