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If all three zeroes of a cubic polynomial $x^{3} + a x^{2} + bx + c$ are positive, then which of the following is correct about $a, b$ and c?

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c must be positive

b must be negative

b must be positive

a must be positive

answer is A.

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Detailed Solution

Given, all three zeroes of a cubic polynomial

x^{3} + a x^{2} + bx + c

are positive
Let us assume that the roots of the equation

x^{3} + a x^{2} + bx + c

=0 is

α, β, γ .

So, sum of roots,

α + β + γ = - a

.
Thus,

a

is negative.
Product of roots, taken two at a time,

αβ + βγ + γα = b

.
Thus,

b

is positive.
Product of roots,

αβγ = - c

.
Thus,

c

is negative.
Therefore,

a

and

c

have negative sign, but

b

has positive sign.
Hence, the correct option is 1.

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If all three zeroes of a cubic polynomial x3+ax2+bx+c are positive, then which of the following is correct about a,b and c?

Detailed Solution

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If all three zeroes of a cubic polynomial $x^{3} + a x^{2} + bx + c$ are positive, then which of the following is correct about $a, b$ and c?