Choose the correct answer from the given four options in the following questions:
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
(A) has no linear term and the constant term is negative.
(B) has no linear term and the constant term is positive.
(C) can have a linear term but the constant term is negative.
(D) can have a linear term but the constant term is positive.

Given: 

One of the zeroes of a quadratic polynomial of the form \( x^{2}+a x+b \) is the negative of the other

To do: 

We have to choose the correct option.

Solution:

Let $p(x) = x^2 + ax+ b$

One of the zeroes of a quadratic polynomial of the form \( x^{2}+a x+b \) is the negative of the other.

Therefore,

Let the zeroes of $p(x)$ be $\alpha$ and $-\alpha$.

Sum of the zeroes $= \alpha+(-\alpha) = -\frac{a}{1}$

$\alpha-\alpha = -a$

$a=0$

Product of zeroes $= \alpha \times (- \alpha) = \frac{b}{1}$

$-\alpha^2= b$ which is possible only when $b < 0$.

Hence, \( x^{2}+a x+b \) has no linear term and the constant term is negative.