Question

If all the zeroes of cubic polynomial x³ + ax² + bx + c are negative then a, b and c all have ______ sign.

Answer 1

If all the zeroes of cubic polynomial x³ + ax² + bx + c are negative then a, b and c all have positive sign.

Explanation:

Let the roots be α, β, γ < 0. By Viete’s formulas for x³ + ax² + bx + c, α + β + γ = –a, αβ + βγ + γα = b and αβγ = –c. Since α + β + γ < 0, we have –a < 0 ⇒ a > 0. Each pairwise product αβ, etc. is positive, so b > 0. The product of three negatives is negative, so αβγ < 0 ⇒ –c < 0 ⇒ c > 0. Hence a, b and c are all positive.