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

If all three zeroes of a cubic polynomial x^{3} + qx^{2} - bx + c are positive, then at least one of a, b and c is non - negative.

If all three zeroes of a cubic polynomial x^{3} + qx^{2} - 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) = ax^{3} + bx^{2} + cx + d, where α, β, γ > 0

Product of all the zeroes = - (constant term) ÷ coefficient of x^{3}

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 x^{3}

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 x^{2}) ÷ coefficient of x^{3}

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