Are the following statements True or False Justify your answers.
If all three zeroes of a cubic polynomial `x^(3)+ax^(2)-bx+c` are positive ,then at least one of a ,b and c is non -negative .

Answer

To determine whether the statement "If all three zeroes of a cubic polynomial \(x^3 + ax^2 - bx + c\) are positive, then at least one of \(a\), \(b\), and \(c\) is non-negative" is true or false, we will analyze the polynomial and its zeroes step by step.

### Step 1: Understanding the Polynomial
The given polynomial is:
\[ P(x) = x^3 + ax^2 - bx + c \]
where the zeroes are denoted as \(\alpha\), \(\beta\), and \(\gamma\).

### Step 2: Applying Vieta's Formulas
According to Vieta's formulas for a cubic polynomial, we know:
1. The sum of the zeroes:
\[ \alpha + \beta + \gamma = -a \]
2. The sum of the products of the zeroes taken two at a time:
\[ \alpha\beta + \beta\gamma + \gamma\alpha = -b \]
3. The product of the zeroes:
\[ \alpha\beta\gamma = -c \]

### Step 3: Analyzing the Conditions
Since we are given that all three zeroes \(\alpha\), \(\beta\), and \(\gamma\) are positive, we can derive the following:

1. **Sum of the Zeroes**:
- Since \(\alpha\), \(\beta\), and \(\gamma\) are positive, their sum \(\alpha + \beta + \gamma\) is also positive.
- Therefore, \(-a\) must be positive, which implies:
\[ a < 0 \]

2. **Sum of the Products of the Zeroes**:
- The expression \(\alpha\beta + \beta\gamma + \gamma\alpha\) is positive since it is a sum of products of positive numbers.
- Hence, \(-b\) must be positive, which implies:
\[ b < 0 \]

3. **Product of the Zeroes**:
- The product \(\alpha\beta\gamma\) is positive since it is the product of three positive numbers.
- Therefore, \(-c\) must be positive, which implies:
\[ c < 0 \]

### Step 4: Conclusion
From the analysis above, we have established that:
- \(a < 0\)
- \(b < 0\)
- \(c < 0\)

This means that all three coefficients \(a\), \(b\), and \(c\) are negative. Thus, the statement "at least one of \(a\), \(b\), and \(c\) is non-negative" is **False**.

### Final Answer
The statement is **False** because all three coefficients \(a\), \(b\), and \(c\) can be negative when all zeroes of the polynomial are positive.

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