If `a, b, c, d` are real numbers such that `(3a + 2b)/(c+d)+3/2=0` then the equation `ax^3 + bx^2 + cx + d =0` has

Options

A. At least one root in `[-2,0]`
B. At least one root in `[0,2]`
C. At least two root in [-2,2]
D. No root in [-2,2]

Answer

To solve the problem step by step, we start with the given equation and analyze it systematically.

### Step 1: Analyze the Given Equation
We have the equation:

\[ \frac{3a + 2b}{c + d} + \frac{3}{2} = 0 \]

### Step 2: Rearranging the Equation
To isolate the fraction, we can rearrange the equation:

\[ \frac{3a + 2b}{c + d} = -\frac{3}{2} \]

### Step 3: Cross Multiplication
Cross-multiplying gives us:

\[ 3a + 2b = -\frac{3}{2}(c + d) \]

### Step 4: Simplifying the Equation
Multiplying both sides by 2 to eliminate the fraction:

\[ 6a + 4b = -3(c + d) \]

### Step 5: Rearranging to Form a Polynomial
Now, we can express the polynomial \( ax^3 + bx^2 + cx + d = 0 \). We need to analyze the behavior of this polynomial.

### Step 6: Finding the Derivative
The derivative of the polynomial is:

\[ f'(x) = 3ax^2 + 2bx + c \]

### Step 7: Evaluating the Function at Specific Points
To find the roots, we can evaluate the function at \( x = 0 \) and \( x = 2 \):

1. **At \( x = 0 \)**:
\[ f(0) = d \]

2. **At \( x = 2 \)**:
\[ f(2) = a(2^3) + b(2^2) + c(2) + d = 8a + 4b + 2c + d \]

### Step 8: Setting Conditions for Roots
For the polynomial to have at least one root in the interval \( [0, 2] \), we need to check the signs of \( f(0) \) and \( f(2) \). If \( f(0) \) and \( f(2) \) have opposite signs, then by the Intermediate Value Theorem, there is at least one root in the interval.

### Step 9: Conclusion
Since we have established that \( f(0) = d \) and \( f(2) = 8a + 4b + 2c + d \), we can conclude that if \( d \) and \( 8a + 4b + 2c + d \) are of opposite signs, then the polynomial \( ax^3 + bx^2 + cx + d = 0 \) has at least one root in the interval \( [0, 2] \).