Inverse z-transform, ROC |z| < 0.4

Step 1. Factor the denominator. z² − 0.1z − 0.2 = (z − 0.5)(z + 0.4). Poles at z = 0.5 and z = −0.4.

Step 2. Improper fraction, so divide first (numerator degree 3 > denominator degree 2):

(2z³ + 2z² + 3z + 5) ÷ (z² − 0.1z − 0.2) = 2z + 2.2, remainder 3.62z + 5.44.

X(z) = 2z + 2.2 + (3.62z + 5.44)/((z − 0.5)(z + 0.4))

Step 3. Partial fractions of the remainder term:

(3.62z + 5.44)/((z − 0.5)(z + 0.4)) = A/(z − 0.5) + B/(z + 0.4)

• A = (3.62·0.5 + 5.44)/(0.5 + 0.4) = 7.25/0.9 = 145/18 ≈ 8.056
• B = (3.62·(−0.4) + 5.44)/(−0.4 − 0.5) = 3.992/(−0.9) = −998/225 ≈ −4.436

Step 4. Invert with ROC |z| < 0.4. Both poles are outside this ROC, so every term is left-sided (anticausal). Use

−aⁿ u[−n−1] ↔ 1/(1 − az⁻¹), |z| < |a|, and the delay rule 1/(z − a) = z⁻¹/(1 − az⁻¹) ↔ −a^(n−1) u[−n].

• 2z ↔ 2δ[n + 1]; 2.2 ↔ 2.2δ[n]
• A/(z − 0.5) ↔ −A (0.5)^(n−1) u[−n]
• B/(z + 0.4) ↔ −B (−0.4)^(n−1) u[−n] = +(998/225)(−0.4)^(n−1) u[−n]

Answer:

x[n] = 2δ[n+1] + 2.2δ[n]
       − (145/18)(0.5)^(n−1) u[−n]
       + (998/225)(−0.4)^(n−1) u[−n]

Check (expand X(z) in powers of z): X(z) = −25 − 2.5z − 133.75z² + 44.375z³ + …, so x[0] = −25, x[−1] = −2.5, x[−2] = −133.75, x[−3] = 44.375. The formula gives x[0] = 2.2 − 27.2 = −25, x[−1] = 2 − 4.5 = −2.5, x[−2] = −133.75, x[−3] = 44.375. ✓ x[n] = 0 for n ≥ 1.