Question:

<p><strong>Step 1:</strong> For a cubic polynomial az³ + bz² + cz + d = 0 with all roots having negative real parts, we apply the Routh-Hurwitz criterion. The necessary and sufficient conditions are that certain determinants are positive (or follow a sign pattern).</p><p><strong>Step 2:</strong> For a cubic az³ + bz² + cz + d = 0, if all roots have negative real parts, then: the polynomial can be written as a(z - r₁)(z - r₂)(z - r₃) where Re(rᵢ) < 0 for all i.</p><p><strong>Step 3:</strong> Expanding this product with Re(rᵢ) < 0 forces constraints on coefficients. Without loss of generality, assume a > 0 (we can multiply by -1). Then all coefficients must be positive: a > 0, b > 0, c > 0, d > 0.</p><p><strong>Step 4:</strong> However, if a < 0 instead, the polynomial becomes -|a|z³ - |b|z² - |c|z - |d| = 0, which means b, c, d < 0 while a < 0. The key observation is that b, c, d must always have the same sign (all positive if a > 0, all negative if a < 0).</p><p><strong>Step 5:</strong> By Routh-Hurwitz stability criterion for cubic equations: if all roots have negative real parts, then a/b > 0, b/c > 0, c/d > 0. This means b, c, d must have the same sign.</p><p><strong>∴ Answer: D</strong></p>