If one of the zeroes of a quadratic polynomial of the form

Solution:

(a) Let      p(x) = x2 + ax + b.

Put a = 0, then,                            p(x) = x2 + b = 0

⇒                                              x2 = -b

⇒                                              x = ±±−b−−−√

[∴b < 0]

Hence, if one of the zeroes of quadratic polynomial p(x) is the negative of the other, then it has no linear term i.e., a = O and the constant term is

negative i.e., b< 0.

Alternate Method

Let                    f(x) = x2 + ax+ b

and by given condition the zeroes area and – α.

Sum of the zeroes = α- α = a

=>a = 0

f(x) = x2 + b, which cannot be linear,

and product of zeroes = α .(- α) = b

⇒                         -α2 = b

which is possible when, b < 0.

Hence, it has no linear term and the constant term is negative.