Class 10 Math All topics Polynomials

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.

(a)

True

(b)

False

Correct answer: (a)

Solution:

Let $p(x)=x_{3}+ax_{2}+bx+c$ is a cubic polynomial and $α,β,γ$ be the roots of $p(x)$

Then, sum of the roots is

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

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 $αβγ=−c$ and product of three negative numbers is negative

Product of the roots is $αβγ=−c$ 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.Therefore sign of all coefficients will be positive.

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