Question

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

Options

• Has no linear term and the constant term is negative

• Has no linear term and the constant term is positive

• Can have a linear term but the constant term is negative

• Can have a linear term but the constant term is positive

Answer 1

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.