Question

Say true or false.

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.

• A. True
• B. False

Answer 1

The correct option is A True

Let $p\left(x\right)=x^3+a{x}^2+bx+c$ is a cubic polynomial and $\alpha ,\beta ,\gamma$ be the roots of $p\left(x\right)$
Then, sum of the roots is

$\alpha +\beta +\gamma =-a$ and sum of negative numbers is negative

Thus $a$ is positive
Product of the roots taken two at a time is

$\alpha .\beta +\alpha .\gamma +\gamma .\beta =b$

Product of two negative numbers is positive and sum of positive numbers is positive

Thus, $b$ is positive
Product of the roots is $\alpha \beta \gamma =-c$ and product of three negative numbers is negative

Thus $c$ is positive.
Therefore sign of all coefficients will be positive.

Hence, the Statement is true.