If `a,b,c,d in R` and all the three roots of `az^3 + bz^2 + cZ + d=0` have negative real parts, then

Options

A. `ab gt 0`
B. `bc gt 0`
C. `ad gt 0`
D. `bc-ad gt 0`

Explanation

To solve the problem, we need to analyze the cubic polynomial \( az^3 + bz^2 + cz + d = 0 \) under the condition that all three roots have negative real parts. Let's break down the solution step by step.

### Step 1: Write the polynomial and its derivative
The given polynomial is:
\[ f(z) = az^3 + bz^2 + cz + d \]
We differentiate this polynomial with respect to \( z \):
\[ f'(z) = 3az^2 + 2bz + c \]

### Step 2: Analyze the critical points
To find the critical points (where the function changes direction), we set the derivative equal to zero:
\[ 3az^2 + 2bz + c = 0 \]
This is a quadratic equation in \( z \).

### Step 3: Apply the quadratic formula
Using the quadratic formula \( z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), we identify:
- \( A = 3a \)
- \( B = 2b \)
- \( C = c \)

Thus, the roots of the derivative are:
\[ z = \frac{-2b \pm \sqrt{(2b)^2 - 4 \cdot 3a \cdot c}}{2 \cdot 3a} \]
This simplifies to:
\[ z = \frac{-2b \pm \sqrt{4b^2 - 12ac}}{6a} \]

### Step 4: Analyze the conditions for the roots
For the original polynomial \( f(z) \) to have all roots with negative real parts, we need the critical points (roots of \( f'(z) \)) to also have negative real parts. This implies:
\[ \frac{-2b}{6a} < 0 \]
This leads to the conclusion that \( a \) and \( b \) must have the same sign.

### Step 5: Check the sign of \( c \)
Next, we consider the value of \( c \). Since the cubic polynomial has three roots with negative real parts, the sum of the roots (which is given by \( -\frac{b}{a} \)) must also be negative. This means \( b \) and \( a \) must have the same sign.

### Step 6: Conclusion about the signs of \( a, b, c, d \)
Since \( a \) and \( b \) have the same sign, we can also conclude that \( c \) must have the same sign as \( a \) and \( b \). The constant term \( d \) can be analyzed similarly, but it will depend on the specific values of the roots.

Thus, we conclude that:
- \( a, b, c \) must have the same sign.

### Final Answer
All coefficients \( a, b, c \) must have the same sign.

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