If $\sqrt{2} & - \sqrt{2}$ are zeroes of $x^{4} - 11 x^{2} + 18$ find the other zeroes. [3 MARKS]

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Solution

Concept : 1 Mark
Application : 1 Mark
Calculation : 1 Mark

Given polynomial is,

$f (x) = x^{4} - 11 x^{2} + 18$

Let $x^{2} = y$

Now $f (y) = y^{2} - 11 y + 18$

$\Rightarrow f (y) = y^{2} - 9 y - 2 y + 18$

$\Rightarrow f (y) = (y - 9) (y - 2)$

For zeros,

$(y - 9) = 0 \Rightarrow y = 9$

$(y - 2) = 0 \Rightarrow y = 2$

$\Rightarrow x^{2} = 9 o r x^{2} = 2$

$\Rightarrow x = \pm 3 o r x = \pm \sqrt{2}$

$∴$ the other roots are $\pm 3$

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