If one of the zeroes of a quadratic polynomial of the form x2 + ax + b is the negative of the other, then it ______.

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If one of the zeroes of a quadratic polynomial of the form x2 + ax + b is the negative of the other, then it has no linear term and the constant term is negative. Explanation: Let p(x) = x2 + ax + b Put a = 0, then, p(x) = x2 + b = 0 ⇒ x2 = – b ⇒ x = +- sqrt(-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 = 0 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.

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