Obtain all other zeroes of 3x^4+6x^32x^210x5, if two of its zeroes are √(5/3) and – √(5/3).
Obtain all other zeroes of 3x^{4}+6x^{3}2x^{2}10x5, if two of its zeroes are √(5/3) and – √(5/3).
Since this is a polynomial equation of degree 4, hence there will be total 4 roots.
√(5/3) and – √(5/3) are zeroes of polynomial f(x).
∴ (x –√(5/3)) (x+√(5/3) = x^{2}(5/3) = 0
(3x^{2}−5) = 0, is a factor of given polynomial f(x).
Now, when we will divide f(x) by (3x^{2}−5) the quotient obtained will also be a factor of f(x) and the remainder will be 0.
Therefore, 3x^{4 }+6x^{3 }−2x^{2 }−10x–5 = (3x^{2 }–5)(x^{2}+2x+1)
Now, on further factorizing (x^{2}+2x+1) we get,
x^{2}+2x+1 = x^{2}+x+x+1 = 0
x(x+1)+1(x+1) = 0
(x+1)(x+1) = 0
So, its zeroes are given by: x = −1 and x = −1.
Therefore, all four zeroes of given polynomial equation are:
√(5/3), √(5/3), −1 and −1.

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