Are the following statements True or False Justify your answers.
If all the zeroes of a cubic polynomial are negative , then all the coefficients and the constant term of the polynomial have the same sign.

Answer

To determine whether the statement "If all the zeroes of a cubic polynomial are negative, then all the coefficients and the constant term of the polynomial have the same sign" is true or false, we can analyze the properties of cubic polynomials and their roots.

### Step-by-Step Solution:

1. **Define the cubic polynomial**:
A cubic polynomial can be expressed in the form:
\[ P(x) = ax^3 + bx^2 + cx + d \]
where \(a\), \(b\), \(c\), and \(d\) are coefficients.

2. **Identify the roots**:
Let the roots (zeroes) of the polynomial be \(\alpha\), \(\beta\), and \(\gamma\). According to the statement, all these roots are negative:
\[ \alpha < 0, \quad \beta < 0, \quad \gamma < 0 \]

3. **Use Vieta's formulas**:
Vieta's formulas relate the coefficients of the polynomial to the sums and products of its roots:
- The sum of the roots:
\[ \alpha + \beta + \gamma = -\frac{b}{a} \]
- The sum of the products of the roots taken two at a time:
\[ \alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a} \]
- The product of the roots:
\[ \alpha\beta\gamma = -\frac{d}{a} \]

4. **Analyze the sum of the roots**:
Since \(\alpha\), \(\beta\), and \(\gamma\) are all negative, their sum \(\alpha + \beta + \gamma\) is also negative. Therefore, from Vieta's first formula:
\[ -\frac{b}{a} < 0 \implies \frac{b}{a} > 0 \]
This implies that \(b\) and \(a\) must have the same sign.

5. **Analyze the sum of the products of the roots**:
The expression \(\alpha\beta + \beta\gamma + \gamma\alpha\) involves products of negative numbers. The product of two negative numbers is positive, and since there are three such products, the entire expression is positive:
\[ \frac{c}{a} > 0 \implies c \text{ and } a \text{ have the same sign.} \]

6. **Analyze the product of the roots**:
The product \(\alpha\beta\gamma\) is negative (since it is the product of three negative numbers):
\[ -\frac{d}{a} < 0 \implies \frac{d}{a} > 0 \]
This implies that \(d\) and \(a\) must have the same sign.

7. **Conclusion**:
From the above analysis, we have established that:
- \(a\) and \(b\) have the same sign.
- \(a\) and \(c\) have the same sign.
- \(a\) and \(d\) have the same sign.

Therefore, all coefficients \(a\), \(b\), \(c\), and constant term \(d\) must have the same sign.

Thus, the statement is **True**.