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प्रश्न
Find all zeros of the polynomial f(x) = 2x4 − 2x3 − 7x2 + 3x + 6, if its two zeros are `-sqrt(3/2)` and `sqrt(3/2)`
उत्तर
Since `-sqrt(3/2)` and `sqrt(3/2)` are two zeros of f(x) Therefore,
`=(x-sqrt(3/2))(x+sqrt(3/2))`
`=(x^2-3/2)`
`=1/2(2x^2-3)` is a factor of f(x).
Also 2x2 - 3 is a factor of f(x).
Let us now divide f(x) by 2x2 - 3. we have
By using that division algorithm we have,
f(x) = g(x) x q(x) + r(x)
2x4 − 2x3 − 7x2 + 3x + 6 = (2x2 - 3)(x2 - x - 2) + 0
2x4 − 2x3 − 7x2 + 3x + 6 `=(sqrt2x+sqrt3)(sqrt2x-sqrt3)(x^2+1x-2x-2)`
2x4 − 2x3 − 7x2 + 3x + 6 `=(sqrt2x+sqrt3)(sqrt2x-sqrt3)[x(x+1)-2(x+1)]`
2x4 − 2x3 − 7x2 + 3x + 6 `=(sqrt2x+sqrt3)(sqrt2x-sqrt3)(x-2)(x+1)`
Hence, The zeros of f(x) are `-sqrt(3/2)`, `sqrt(3/2)`, 2 , -1.
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