State whether the following statement is 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.

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True

False

answer is A.

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Detailed Solution

Gicen that all the zeroes of a cubic polynomial are negative.
We know that, cubic polynomials have at most three real zeroes.
Graph of the cubic polynomial neither touches nor intersects the x-axis.
Let,

a x^{3} + b x^{2} + cx + d

be the cubic polynomial and

α, β, and γ

are the zeroes of the polynomial and

α, β, and γ

are negative.
Then,

α + β + γ = \frac{- b}{a}

The sum of three negative number always gives a negative value, which satisfies the above equation.
And

αβ + βγ + γα = \frac{c}{a}

The product of two negative number is positive, which satisfy the above equation.
And

αβγ = \frac{- d}{a}

The product of three negative numbers gives a negative value which satisfies the above equation.
Therefore, the given statement is true.
Hence, option (1) is correct.

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