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Inverse z of (2z²-5z)/((z-2)(z-3))

Short answer

Divide by z and use partial fractions: X(z)=z/(z-2)+z/(z-3). Inverse gives x(k)=2^k+3^k for k≥0.

Find the inverse z-transform of (2z² - 5z)/((z - 2)(z - 3)) by using partial fraction method.

Partial fractions. Divide by z first, because every standard pair has a factor z in the numerator.

X(z)/z = (2z − 5)/[(z − 2)(z − 3)] = A/(z − 2) + B/(z − 3)

A = (2·2 − 5)/(2 − 3) = (−1)/(−1) = 1
B = (2·3 − 5)/(3 − 2) = 1/1       = 1

So X(z) = z/(z − 2) + z/(z − 3).

Using Z⁻¹[z/(z − a)] = aᵏ:

Answer: x(k) = 2ᵏ + 3ᵏ, k ≥ 0.

Check: x(0) = 2 (= lim X(z)), x(1) = 5, x(2) = 13, x(3) = 35 — matches the power-series expansion of X(z).

✓ Answer checked by Claude Opus 5.5 (high effort)

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