If one of the zeroes of a quadratic polynomial of the form x2 + ax + b is the negative of the other, then it
let p(x) = x2 + ax + b
And let α be one of the zeroes
∴ - α is the other zero of the polynomial p(x)
Product of the zeroes = constant term ÷ coefficient of x2
Product of the zeroes = b
α(- α) = b
- α2 = b
i.e. b is negative.
Sum of the zeroes = - (coefficient of x) ÷ coefficient of x2
α - α = - a
0 = - a
⇒ a = 0
∴ The polynomials can be written as x2 - α2 (No linear term and negative constant term)