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Obtain all zeros of the polynomial (2x^...

Obtain all zeros of the polynomial `(2x^(3)-4x-x^(2)+2),` if two of its zeros are ` sqrt2 and (-sqrt2.`

Text Solution

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The given polynomial is `f(x) = 2x^(3) - x^(2) - 4x+2.`
Since `sqrt2 and -sqrt2` are the zeros of f(x), it follows that each one of `(x-sqrt2) and (x+sqrt2)` is a factor of f(x).
Consequently , `(x-sqrt2)(x+sqrt2) = (x^(2)-2)` is a factor of f(x).
On dividing `f(x) = 2x^(3)-x^(2)-4x+2" by "(x^(2)-2)`, we get

`:. f(x) = 0 rArr (x^(2)-1) (2x-1) = 0`
` rArr (x-sqrt2)(x+sqrt2)(2x-1) = 0`
` rArr (x-sqrt2)=0 or (x+sqrt2) =0 or (2x-1) =0 `
` rArr x = sqrt2 or x =- sqrt2 or x = 1/2.`
Hence, all zeros of f(x) are ` sqrt2, -sqrt2 and 1/2.`
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