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Question
Find a quadratic polynomial with the given numbers as the sum and product of its zeroes, respectively.
`sqrt2 , 1/3`
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Solution
Given: α + β = `sqrt2`, αβ = `1/3`
Since ax2 + bx + c = kx2 - k(α + β)x + kαβ
or ax2 + bx + c = k[x2 - (α + β)x + αβ]
Or `(ax^2 + bx + c)/k = (x^2 - sqrt2x + 1/3)`
Or `(ax^2 + bx + c)/k = (3x^2 - 3sqrt2x + 1)/3`
Here k is a constant term, by comparing k = 3
Hence, ax2 + bx + c = `3x^2 - 3sqrt2x + 1`
The quadratic polynomial is `3x^2 - 3sqrt2x + 1`.
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