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प्रश्न
Obtain all zeros of the polynomial f(x) = x4 − 3x3 − x2 + 9x − 6, if two of its zeros are `-sqrt3` and `sqrt3`
उत्तर
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) = x4 − 3x3 − x2 + 9x − 6 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)
x4 − 3x3 − x2 + 9x − 6 = (x2 - 3)(x2 - 3x + 2) + 0
x4 − 3x3 − x2 + 9x − 6 = (x2 - 3)(x2 - 2x + 1x + 2)
x4 − 3x3 − x2 + 9x − 6 = (x2 - 3)[x(x - 2) - 1(x - 2)]
x4 − 3x3 − x2 + 9x − 6 = (x2 - 3)[(x - 1)(x - 2)]
x4 − 3x3 − x2 + 9x − 6 `= (x - sqrt3)(x+sqrt3)(x-1)(x-2)`
Hence, the zeros of the given polynomials are `-sqrt3`, `sqrt3`, +1 and +2.
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