Question

1. $(w^3 - 2w^2 - 2w + 1) div (w - 1)$

Explanation:

Step1: Use polynomial long division

Divide the first term of the dividend \(w^3\) by the first term of the divisor \(w\), we get \(w^2\). Multiply the divisor \((w - 1)\) by \(w^2\) to get \(w^3 - w^2\). Subtract this from the dividend:
\[

$$\begin{align*} &(w^3 - 2w^2 - 2w + 1)-(w^3 - w^2)\\ =&w^3 - 2w^2 - 2w + 1 - w^3 + w^2\\ =&-w^2 - 2w + 1 \end{align*}$$

\]

Step2: Divide the first term of the new dividend \(-w^2\) by \(w\)

We get \(-w\). Multiply the divisor \((w - 1)\) by \(-w\) to get \(-w^2 + w\). Subtract this from the new dividend:
\[

$$\begin{align*} &(-w^2 - 2w + 1)-(-w^2 + w)\\ =&-w^2 - 2w + 1 + w^2 - w\\ =&-3w + 1 \end{align*}$$

\]

Step3: Divide the first term of the new dividend \(-3w\) by \(w\)

We get \(-3\). Multiply the divisor \((w - 1)\) by \(-3\) to get \(-3w + 3\). Subtract this from the new dividend:
\[

$$\begin{align*} &(-3w + 1)-(-3w + 3)\\ =&-3w + 1 + 3w - 3\\ =&-2 \end{align*}$$

\]
So, the division gives \(w^2 - w - 3\) with a remainder of \(-2\), or we can write it as \(w^2 - w - 3-\frac{2}{w - 1}\)

Answer:

\(w^2 - w - 3-\frac{2}{w - 1}\) (or \(w^2 - w - 3\) with remainder \(-2\))