Are the following statements True or False Justify your answers.
If the zeroes of a quadratic polynomial `ax^(2)+bx+c` are both positive ,then a ,b and c all have the same sign .

Answer

To determine whether the statement "If the zeroes of a quadratic polynomial \( ax^2 + bx + c \) are both positive, then \( a, b, \) and \( c \) all have the same sign" is true or false, we can analyze the conditions of the quadratic polynomial.

### Step-by-Step Solution:

1. **Understanding the Quadratic Polynomial**:
A quadratic polynomial can be expressed as:
\[ f(x) = ax^2 + bx + c \]
where \( a, b, \) and \( c \) are coefficients.

2. **Roots of the Polynomial**:
Let the roots (zeroes) of the polynomial be \( \alpha \) and \( \beta \). According to Vieta's formulas:
- The sum of the roots is given by:
\[ \alpha + \beta = -\frac{b}{a} \]
- The product of the roots is given by:
\[ \alpha \beta = \frac{c}{a} \]

3. **Condition of Roots**:
Given that both roots \( \alpha \) and \( \beta \) are positive, we can infer the following:
- Since \( \alpha + \beta > 0 \), it implies that \( -\frac{b}{a} > 0 \). This means that \( b \) and \( a \) must have opposite signs (if \( a > 0 \), then \( b < 0\) and vice versa).
- Since \( \alpha \beta > 0 \), it implies that \( \frac{c}{a} > 0 \). This means that \( c \) and \( a \) must have the same sign.

4. **Conclusion on Signs**:
From the above deductions:
- If \( a > 0 \), then \( b < 0 \) and \( c > 0 \) (not all the same sign).
- If \( a < 0 \), then \( b > 0 \) and \( c < 0 \) (again, not all the same sign).

5. **Final Statement**:
Since it is possible for \( a, b, \) and \( c \) to not all have the same sign while still having both roots positive, we conclude that the statement is **False**.

### Justification:
The statement is false because the conditions for the signs of \( a, b, \) and \( c \) do not guarantee that they are all the same when both roots are positive.