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State whether the following statement is true or false.

If all three zeroes of a cubic polynomial $x 3 + a x 2 − b x + c$ are positive, then at least one of a, b and c is non-negative.

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True

False

answer is A.

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Detailed Solution

Given cubic polynomial is

m x 3 + n x 2 + l x + q .

Let’s assume

a, b,

and

c

are three roots of the given cubic polynomial.
Then,

a + b + c = − n m

The sum of three positive numbers always gives a positive value, which satisfies the equation

a + b + c = − n m

if the values of

n and m

have opposite signs.
And

a b + b c + a c = l m

The product of two positive number is always positive, which satisfy the equation

a b + b c + c a = l m,

if the signs of

c and a

are the same.
And

a b c = − q m

The product of three positive numbers gives a positive value which satisfies the equation

a b c = − q m

q and m

have opposite signs.
So, the signs of coefficients must be different from each other.
Therefore, the given statement is true.
Hence, option (1) is correct.

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