If one of the zeroes of a quadratic polynomial of the form `x^2 + ax + b` is the negative of the other, then it

Options

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

A

Explanation

Let `p(x) = x^(2)+ax +b`.
Now, product of zeroess `=("Constant term")/("Coefficient of" x^(2))`
Let `alpha` and `beta` be the zeroes of `p(x)`.
`:.` Product of zeroes `(alpha. beta) = (b)/(1)`
`rArr alpha beta = b` ..(i)
Given that, one of the zeroes of a quadratic polynomial `p(x)` is negative of the other.
`:. alpha beta lt 0`
so, `b lt 0` " "[from Eq.(i)]
Hence, b should be negative
Put `a = 0`, then, `p(x) = x^(2) +b = 0`
`rArr x^(2) =- b`
`rArr x = +- sqrt(-b) [ :' b lt 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 lt 0`.
Alternate Method
Let `f(x) = x^(2) +ax +b`
any by given condition the zeroes are `alpha` and `-alpha`.
`:.` Sum of the zeroes `= alpha - alpha = a`
`rArr a = 0`
`:. f(x) = x^(2)+b`, which cannot be linear.
and product of zeroes `= alpha.(-alpha) = b`
`rArr -alpha^(2) = b`
which is possible when, `b lt 0`.
Hence, it has no linear term and the constant term is negative.