Question

If all three zeroes of a cubic polynomial $x^3+a{x}^2+bx+c$ are positive, then which of the following is correct about $a,b$ and $c$?

• A. $b$ must be positive
• B. $a$ must be positive
• C. $c$ must be positive
• D. $b$ must be negative

Answer 1

The correct option is A $b$ must be positive

Let the roots of the equation $x^3+a{x}^2+bx+c=0$ be $\alpha ,\beta ,\gamma$

Then, sum of roots, $\alpha +\beta +\gamma =-a$.

Thus, $a$ is negative.

Product of roots, taken two at a time, $\alpha .\beta +\alpha .\gamma +\gamma .\beta =b$.

Thus $b$ is positive

Product of roots, $\alpha .\beta .\gamma =-c$.

Thus $c$ is negative.

Hence, $a$ and $c$ have negative sign, but $b$ has positive sign.