Q. 2 E5.0( 4 Votes )

Are the following statements ‘true’ or ‘False’? Justify your answer.

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.

Answer :

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: True

Let α, β and γ be the zeroes of the polynomial p(x) = ax3 + bx2 + cx + d, where α, β, γ < 0


Product of all zeroes = - (constant term) ÷ coefficient of x3


αβγ = - d/a < 0 ( α, β, γ < 0 αβγ < 0)


d/a > 0


d and a have the same signs.


Sum of the products of two zeroes at a time = coefficient of x ÷ coefficient of x3


αβ + βγ + αγ = c/a > 0 ( α, β, γ < 0 αβ,βγ,αγ > 0 αβ + βγ + αγ > 0 )


c and a have the same signs.


Sum of the zeroes = - (coefficient of x2) ÷ coefficient of x3


α + β + γ = - b/a < 0 ( α, β, γ < 0 α + β + γ < 0)


b/a > 0


b and a have same signs.


a,b,c and d have same signs.


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