(z^4 - 3z^2 + 2z^2 - 4z + z)(z - 1)^{-1}

Steps to Solve

1. Combine Like Terms in the Polynomial

First, let's simplify the polynomial in the expression: $$ z^4 - 3z^2 + 2z^2 - 4z + z $$

Combine the terms:

• Combine $-3z^2 + 2z^2$ to get $-z^2$.
• Combine $-4z + z$ to get $-3z$.

So, the simplified polynomial is: $$ z^4 - z^2 - 3z $$

2. Rewrite the Expression

Now, our expression becomes: $$ (z^4 - z^2 - 3z)(z - 1)^{-1} $$

This means we're dividing the polynomial by $(z - 1)$.

3. Factor the Polynomial

Next, let's factor the polynomial $z^4 - z^2 - 3z$.

Notice we can factor out $z$: $$ z(z^3 - z - 3) $$

4. Divide by $(z - 1)$

We now consider: $$ \frac{z(z^3 - z - 3)}{z - 1} $$

If $z - 1$ is a factor of the cubic polynomial (which can be checked using synthetic division), we can simplify further.

5. Check for Factor $(z - 1)$

To check if $(z - 1)$ is a factor of $z^3 - z - 3$, we can substitute $z = 1$: $$ 1^3 - 1 - 3 = 1 - 1 - 3 = -3 \text{ (not a factor)} $$

Instead, we cannot simplify this polynomial by $(z - 1)$.

6. Final Simplified Expression

Therefore, our final expression remains: $$ \frac{z(z^3 - z - 3)}{z - 1} $$