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

let p(x) = x^{2} + 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 x^{2}

Product of the zeroes = b

α(- α) = b

- α^{2} = b

i.e. b is negative.

Sum of the zeroes = - (coefficient of x) ÷ coefficient of x^{2}

α - α = - a

0 = - a

⇒ a = 0

∴ The polynomials can be written as x^{2} - α^{2} (No linear term and negative constant term)

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