Are the following statements ‘true’ or ‘False’? Justify your answer.
If all the zeroes of a cubic polynomial are negative, then all the coefficients and the constant term of the polynomial have the same sign.

Question

Philoid · Accepted Answer

If all the zeroes of a cubic polynomial are negative, then all the coefficients and the constant term of the polynomial have the same sign: 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 zeroes = - (constant term) ÷ coefficient of x³

αβγ = - d/a < 0 (∵ α, β, γ < 0 ⇒ αβγ < 0)

⇒ d/a > 0

⇒ d and a have the same 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 same signs.

⇒ a,b,c and d have same signs.

Are the following statements ‘true’ or ‘False’? Justify your answer.If all the zeroes of a cubic polynomial are negative, then all the coefficients and the constant term of the polynomial have the same sign.