Is the following statement True or False? Justify your answer.If all t

1.	Is the following statement 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.
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: 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.

Is the following statement 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.

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: 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.