Inverse z-transform of factored X(z)

X(z) = z²(1 − (3/2)z⁻¹)(1 + z⁻¹)(1 − z⁻¹) is a polynomial in z, so expand it; each term cz⁻ᵏ is simply a delayed impulse cδ[n − k].

(1 + z⁻¹)(1 − z⁻¹) = 1 − z⁻²

(1 − 1.5z⁻¹)(1 − z⁻²) = 1 − 1.5z⁻¹ − z⁻² + 1.5z⁻³

Multiply by z²:

X(z) = z² − 1.5z − 1 + 1.5z⁻¹

Use δ[n − k] ↔ z⁻ᵏ (k negative means an advance):

• z² ↔ δ[n + 2]
• −1.5z ↔ −1.5δ[n + 1]
• −1 ↔ −δ[n]
• 1.5z⁻¹ ↔ 1.5δ[n − 1]

Answer: x[n] = δ[n + 2] − 1.5δ[n + 1] − δ[n] + 1.5δ[n − 1],

i.e. x[n] = {1, −1.5, −1, 1.5} for n = −2, −1, 0, 1 (zero elsewhere). It is a finite-length sequence, so the ROC is the whole z-plane except z = 0 and z = ∞.