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Question
Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
`1/4 , -1`
Solution
Given: α + β = `1/4`, αβ = -1
Since ax2 + bx + c = kx2 - k(α + β)x + kαβ
In comparison,
a = k, b = -k(α + β) and c = kαβ
α + β = `(-b)/a = 1/4` and αβ = `c/a = -1`
⇒ a = 4
⇒ b = -4(α + β)
⇒ c = kαβ = 4(-1)
Hence, on writing as ax2 + bx + c
⇒ 4x2 - 4(α + β)x + 4(αβ)
⇒ `4x^2 - 4(1/4)x + 4(-1)`
⇒ 4x2 - x - 4
The quadratic polynomial is 4x2 - x - 4.
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