Are the following statements 'True' or 'False'? Justify your answers.
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.

Given:

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.

To do:

We have to find whether the given statement is true or false.

Solution:

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

This implies,

Sum of the roots $=\alpha+\beta+\gamma=-a$

Sum of negative numbers is negative.

This implies,

$a$ is positive.
Product of the roots taken two at a time $=\alpha \cdot \beta+\alpha \cdot \gamma+\gamma \cdot \beta=b$

Product of two negative numbers is positive and sum of positive numbers is positive.
This implies,

$b$ is positive
Product of the roots $=\alpha \beta \gamma=-c$

Product of three negative numbers is negative
This implies,

$c$ is positive.
Therefore, the sign of all three coefficients will be positive.
Hence, the given statement is true.

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Updated on: 10-Oct-2022

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