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Zeros of a Polynomial

If all the ze...

Question

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

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Solution

Let f(x) = x³ + ax²– bx + c

$Let the zeroes of f (x) be α, β, γ, where all these zeroes are negative . Then, α + β + γ = - a Thus, a is positive (∵ α, β, γ are negative) α β + β γ + γ α = - b Thus, b is positive (∵ α, β, γ are negative) α β γ = - c Thus, c is positive (∵ α, β, γ are negative)$

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

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