Question

Is the following statement true or false? Justify your answers.
If all the three zeroes of a cubic polynomial $x^3+a{x}^2-bx+c$ are positive, then at least one of a, b and c is non-negative.

• A. True
• B. False

Answer 1

The correct option is B False

Let

$\alpha ,\beta$ and $\gamma$ are all $+\text{ve}$ for cubic polynomial $x^3+a{x}^2+bx+c.$

Then

$\alpha +\beta +\gamma =-a$ ...(i)

$\alpha \beta +\beta \gamma +\gamma \alpha =b$ ...(ii)

$\alpha \beta \gamma =-c$ ...(iii)

$\because \alpha ,\beta ,\gamma$ are all positive so,

$\alpha +\beta +\gamma >0$

$\Rightarrow -a>0$ [From (i)]

$\Rightarrow a<0$

$\alpha \beta +\beta \gamma +\gamma \alpha >0$

$\Rightarrow b>0$ [From (ii)]

And, $\alpha \beta \gamma >0$

$\Rightarrow -c>0$ [From (ii) ]

$\Rightarrow c<0$

Hence $a,c$ are negative while $b$ is positive.