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) = ax3 + bx2 + 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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