Is the following statement True or False? Jus | vedclass

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(A) The statement is True.Let the zeroes of the quadratic polynomial $p(x) = ax^2 + bx + c$ be $\alpha$ and $\beta$,where $\alpha < 0$ and $\beta < 0$

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True

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(A) The statement is True.Let the zeroes of the quadratic polynomial $p(x) = ax^2 + bx + c$ be $\alpha$ and $\beta$,where $\alpha From the relationship between zeroes and coefficients:$1$. The sum of the zeroes is $\alpha + \beta = -\frac{b}{a}$. Since both $\alpha$ and $\beta$ are negative,their sum $(\alpha + \beta)$ must be negative. Therefore,$-\frac{b}{a}  0$. This means $a$ and $b$ must have the same sign.$2$. The product of the zeroes is $\alpha \cdot \beta = \frac{c}{a}$. Since the product of two negative numbers is positive,$\frac{c}{a} > 0$. This means $a$ and $c$ must have the same sign.Since $a$ and $b$ have the same sign,and $a$ and $c$ have the same sign,it follows that $a, b,$ and $c$ all have the same sign.

Is the following statement True or False? Justify your answer. If the zeroes of a quadratic polynomial $ax^2 + bx + c$ are both negative,then $a, b,$ and $c$ all have the same sign.

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