The Total solution for NCERT class 6-12
If one of the zeroes of a quadratic polynomial of the form x2 + ax + b is the negative of the other, then it(a) has no linear term and constant term isnegative.(b) has no linear term and the constant term ispositive.(c) can have a linear term but the constant termis negative.(d) can have a linear term but the constant termis positive.
(a)
Given that, one of the zeroes of a quadratic polynomial p(x)is negative of the other.αβ < 0So, b < 0 [from Eq. (i)]Hence, b should be negative Put a = 0, then,p(x) = x2 + b = 0 => x2 = – b=> x = ± √-b [ b < 0]Hence, if one of the zeroes of quadraticpolynomial 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 Letf(x) = x2 + ax + b and by given condition the zeroes are a and-a. Sum of the zeroes = α – α = a => a = 0f(x) = x2 + b, which cannot be linear and product of zeroes = α(-α) = b=> – α2 = bwhich is possible when, b < 0.Hence, it has no linear term and the constant tennis negative.