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Nature of Roots

Find all the ...

Question

Find all the zeros of the polynomial $2 x^{4} - 3 x^{3} - 3 x^{2} + 6 x - 2$ if you know that two of its zeros are $\sqrt{2}$ and $- \sqrt{2}$ .

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Solution

Given $\sqrt{2}$ and $- \sqrt{2}$ are two roots
$\Rightarrow x^{2} - 2 = 0$
$x^{2} - 2) ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ 2 x^{4} - 3 x^{3} - 3 x^{2} + 6 x - 2 (2 x^{2} - 3 x + 1 (-) 2 x^{4} - 4 x^{2}- 3 x^{3} {+ x}^{2} + 6 x - 2 - {3 x}^{3} + 6 x (-) (-)x^{2} - 2 x^{2} - 20$

$∴ 2 x^{2} - 3 x + 1 = 0$ is other equation having two roots
$i . e, 2 x^{2} - 2 x - x + 1 = 0$
$2 x (x - 1) - 1 (x - 1) = 0$
$x = 1 / 2$ and $x = 1$

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