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# 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.

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