If a,b,c d `in R` such that `(a+2c)/(b+3d) +4/3=0,` then the equation `ax ^(3) + bx^(3)+ cx+d =0` has

Options

A. atleast one root in `(-1, 0)`
B. atleast one root in `(0,1)`
C. no root in `(-1, 1)`
D. no root in `(0,2)`

Answer

To solve the problem step-by-step, we need to analyze the given equation and the condition provided.

### Step 1: Analyze the Given Condition
We start with the condition:
\[ \frac{a + 2c}{b + 3d} + \frac{4}{3} = 0 \]
This can be rewritten as:
\[ \frac{a + 2c}{b + 3d} = -\frac{4}{3} \]
Cross-multiplying gives:
\[ 3(a + 2c) + 4(b + 3d) = 0 \]
Expanding this results in:
\[ 3a + 6c + 4b + 12d = 0 \]

### Step 2: Rearranging the Equation
We can rearrange the equation to isolate terms involving \(a\), \(b\), \(c\), and \(d\):
\[ 3a + 4b + 6c + 12d = 0 \]

### Step 3: Expressing \(d\) in Terms of \(a\), \(b\), and \(c\)
From the equation \(3a + 4b + 6c + 12d = 0\), we can express \(d\) as:
\[ 12d = -3a - 4b - 6c \]
\[ d = -\frac{3a + 4b + 6c}{12} \]

### Step 4: Considering the Polynomial Equation
Now, we consider the polynomial equation:
\[ ax^3 + bx^2 + cx + d = 0 \]
Substituting \(d\) from the previous step, we have:
\[ ax^3 + bx^2 + cx - \frac{3a + 4b + 6c}{12} = 0 \]

### Step 5: Finding Roots
To determine the nature of the roots of this cubic equation, we can use Rolle's Theorem. We need to find the values of \(f(0)\) and \(f(1)\):
- \(f(0) = d = -\frac{3a + 4b + 6c}{12}\)
- \(f(1) = a + b + c + d = a + b + c - \frac{3a + 4b + 6c}{12}\)

### Step 6: Setting Up for Rolle's Theorem
We need to check if \(f(0) = 0\) and \(f(1) = 0\):
1. If \(f(0) = 0\), then:
\[ -\frac{3a + 4b + 6c}{12} = 0 \implies 3a + 4b + 6c = 0 \]
This is satisfied by our earlier condition.

2. If \(f(1) = 0\), we need to simplify:
\[ a + b + c - \frac{3a + 4b + 6c}{12} = 0 \]

### Step 7: Conclusion Using Rolle's Theorem
Since both \(f(0) = 0\) and \(f(1) = 0\), by Rolle's Theorem, there exists at least one root in the interval \((0, 1)\). Thus, the equation \(ax^3 + bx^2 + cx + d = 0\) has at least one real root.

### Final Answer
The equation \(ax^3 + bx^2 + cx + d = 0\) has at least one real root.

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