To find the inverse Z-transform of the given expression using the long division method, we begin by simplifying the expression and applying the method to determine the inverse. Let's start with the expression:

$\frac{10z^{-1}}{(z-1)(z-2)}$

Step 1: Rewrite the expression.

The given expression can be rewritten as:

$X(z) = \frac{10z}{(z-1)(z-2)}$

Step 2: Perform long division.

Now, we divide $10z$ by $(z-1)(z-2)$. We need to express this as a power series, which will give us the inverse Z-transform in terms of a sequence. First, expand $(z-1)(z-2)$:

$(z-1)(z-2) = z^2 - 3z + 2$

Now, perform long division of $10z$ by $z^2 - 3z + 2$.

1. Divide $10z$ by $z^2$, which gives us $\frac{10}{z}$.

2. Multiply $\frac{10}{z}$ by $z^2 - 3z + 2$, and subtract this from $10z$.

   $\frac{10}{z} \times (z^2 - 3z + 2) = 10z - 30 + \frac{20}{z}$

   Subtract from $10z$, we get:

   $(10z - 10z) + 30 - \frac{20}{z} = 30 - \frac{20}{z}$

Now, divide $30 - \frac{20}{z}$ by $z^2 - 3z + 2$. This process can be continued, but we are now generating a series. Each term generated gives us a term in the Z-transform sequence.

Step 3: Find the inverse Z-transform.

The inverse Z-transform is the sequence whose Z-transform corresponds to the power series we generate from long division.

In this case, the inverse Z-transform will be a series involving terms like $10, -30, 20, \dots$. These terms correspond to the coefficients found in the long division process.

Final Answer:

The inverse Z-transform is the sequence:

$x[n] = 10, -30, 20, \dots$

Would you like to go through the division process step-by-step with more details?

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Next Steps:

1. Would you like to see the complete long division for each step?
2. Do you want to review Z-transform properties?
3. Would you like a recap on how power series relate to Z-transforms?
4. Should we explore a different method of finding the inverse Z-transform?
5. Would you prefer an example with simpler terms?

Tip: The long division method converts the Z-transform into a power series where each term can be easily interpreted as part of the sequence corresponding to the inverse Z-transform.