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Say true or false. If all the zeroes of a cubic polynomial ar

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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.

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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.

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Text solutionVerified

Let

p (x) = x^{3} + a x^{2} + b x + c

is a cubic polynomial and

α, β, γ

be the roots of

p (x)

Then, sum of the roots is

α + β + γ = - a

and sum of negative numbers is negative

Thus

a

is positive
Product of the roots taken two at a time is

α . β + α . γ + γ . β = b

Product of two negative numbers is positive and sum of positive numbers is positive

Thus,

b

is positive
Product of the roots is

α β γ = - c

and product of three negative numbers is negative

Thus

c

is positive.
Therefore sign of all coefficients will be positive.

Hence, the Statement is true.

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Question Text	Say true or false. 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.
Answer Type	Text solution:1
Upvotes	150

Say true or false.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.

Text solutionVerified

Practice questions from similar books

Say true or false.
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.