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State whether the following statement is true or false.

If one of the zeroes of a quadratic polynomial of the form $x 2 + a x + b$ is the negative

of the other, then it has no linear term and the constant term is negative.

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True

False

answer is A.

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Detailed Solution

Given quadratic polynomial is

x 2 + a x + b = 0

and one of the zeroes of the polynomial is the negative of the other.
We know that in quadratic polynomial

m x^{2} + nx + l = 0

,
Sum of zeroes

= \frac{- n}{m}

and product of zeroes

= \frac{l}{m}

.
On comparing given polynomial with standard form we get,

m = 1, n = a, l = b

.
Let

α

be one of the zeroes, then

- α

would be the other zero according to the question.
So, sum of the roots,

α + (- α) = \frac{- a}{1}

\Rightarrow 0 = - a

\Rightarrow - a = 0

Thus, there will be no linear term in the polynomial.
And product of the roots,

α \times (- α) = \frac{b}{1}

\Rightarrow - α^{2} = b

\Rightarrow α^{2} = - b

The product of the root is negative.
Thus, constant term is negative.
Therefore, the given statement is true.
Hence, option 1 is correct.

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