##### 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