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

If all three zeroes of a cubic polynomial x3 + qx2 - bx + c are positive, then at least one of a, b and c is non - negative.


If all three zeroes of a cubic polynomial x3 + qx2 - bx + c are positive, then at least one of a, b and c is non - negative: 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 the zeroes = - (constant term) ÷ coefficient of x3


αβγ = - d/a > 0 ( α, β, γ > 00 αβγ > 0)


d/a < 0


d and a have different 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 different signs.


Case1: when a > 0 c > 0 , b < 0 and d < 0


Case2: when a < 0 c < 0 , b > 0 and d > 0


in both cases two of the coefficients are non - negative.


2
1