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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
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Solution
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.
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