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Question
Find all the zeros of the polynomial 2x3 + x2 − 6x − 3, if two of its zeros are `-sqrt3` and `sqrt3`
Solution
we know that, if x = a is a zero of a polynomial, then x - a is a factor of f(x).
since `sqrt3` and `-sqrt3` are zeros of f(x).
Therefore
`(x+sqrt3)(x-sqrt3)=x^2+sqrt3x-sqrt3x-3`
= x2 - 3
x2 - 3 is a factor of f(x). Now , we divide f(x) = 2x3 + x2 − 6x − 3 by g(x) = x2 - 3 to find the other zeros of f(x).
By using that division algorithm we have,
f(x) = g(x) x q(x) + r(x)
2x3 + x2 − 6x − 3 = (x2 - 3)(2x + 1) + 0
2x3 + x2 − 6x − 3 `= (x+sqrt3)(x-sqrt3)(2x+1)`
Hence, the zeros of the given polynomial are `-sqrt3`, `+sqrt3`, `(-1)/2`.
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