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The coefficient of z^2 in the polynomial (z-1)^3(2z-1)^3 is equal to A. 33 B. -3 C. 30 D. 21

GATE 2023 · General Aptitude · Polynomials · medium

Answer: The coefficient of z^2 in (z-1)^3(2z-1)^3 is 33.

  1. Expand (z-1)^3: (z-1)^3 = z^3 - 3z^2 + 3z - 1. Coefficients: [z^0: -1], [z^1: 3], [z^2: -3], [z^3: 1].
  2. Expand (2z-1)^3: (2z-1)^3 = 8z^3 - 12z^2 + 6z - 1. Coefficients: [z^0: -1], [z^1: 6], [z^2: -12], [z^3: 8].
  3. Extract coefficient of z^2 in the product: [z^2] = [z^0 in first]*[z^2 in second] + [z^1 in first]*[z^1 in second] + [z^2 in first]*[z^0 in second] = (-1)*(-12) + (3)*(6) + (-3)*(-1) = 12 + 18 + 3 = 33.