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Question
Find all zeroes of the polynomial `(2x^4 - 9x^3 + 5x^2 + 3x - 1)` if two of its zeroes are `(2 + sqrt3)` and `(2 - sqrt3)`
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Solution
It is given that `2 + sqrt3` and `2 - sqrt3` are two zeroes of the polynomial f(x) = `24^4 - 9x^3 + 5x^2 + 3x - 1`
`:. {x - (2 + sqrt3)} {x - (2-sqrt3)} = (x - 2 - sqrt3) (x - 2 + sqrt3)`
` = (x - 2)^2 - (sqrt3)^2`
`= x^2 - 4x + 4 - 3`
`= x^2 - 4x + 1`
is a factor of f(x)
Now divide f(x) by `x^2 - 4x + 1`
`:, f(x) = (x^2 - 4x + 1)(2x^2 - x - 1)`
Hence, other two zeroes of f(x) are the zeroes of the polynomial `2x^2 - x - 1`
`2x^2 - x - 1 = 2x^2 - 2x + x - 1 = 2x(x - 1)+ 1(x - 1) = (2x + 1) (x - 1)``
Hence the other two zeroes are `-1/2` and 1
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