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.