Question

If one of the zeroes of a quadratic polynomial of the form x
2+ax + 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.

Answer 1

(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.