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.

Please justify your answer.

Consider a cubic polynomial f(x) = *ax*^{3} + *bx*^{2} + *cx* + *d*. Let -α, -β and -γ be the zeros of f(x).

Now, sum of zeros = -α + (-β) + (-γ) =

⇒ -[α + β + γ] =

⇒ [α + β + γ] = which is positive.

Similarly product of zeros taken two at a time = -α × (-β) + (-β) × (-γ) + (-α)× (-γ) =

⇒ αβ + βγ + αγ = , which is again a positive value.

Now, product of zeros = (-α) ×(-β) × (-γ) =

⇒ -αβγ =

⇒ αβγ = , which is also a positive value.

Hence, 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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