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.

(B) The statement is False.
Let the zeroes of the quadratic polynomial $ax^2 + bx + c$ be $\alpha$ and $\beta$. Since both zeroes are positive,$\alpha > 0$ and $\beta > 0$.
From the relationship between zeroes and coefficients:
$1$. The product of zeroes $\alpha \cdot \beta = \frac{c}{a}$. Since $\alpha > 0$ and $\beta > 0$,their product $\alpha \cdot \beta > 0$. Therefore,$\frac{c}{a} > 0$,which means $a$ and $c$ must have the same sign.
$2$. The sum of zeroes $\alpha + \beta = -\frac{b}{a}$. Since $\alpha > 0$ and $\beta > 0$,their sum $\alpha + \beta > 0$. Therefore,$-\frac{b}{a} > 0$,which implies $\frac{b}{a} < 0$. This means $a$ and $b$ must have opposite signs.
Thus,$a$ and $c$ have the same sign,but $b$ has the opposite sign to $a$ and $c$. Hence,$a, b,$ and $c$ do not all have the same sign.