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How do you factor $x^{4} - x^{2} - 20$ ?

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(x + \sqrt{5}) (x - \sqrt{5}) (x^{2} + 4)

(x - \sqrt{5}) (x - \sqrt{5}) (x^{2} + 4)

(x + \sqrt{5}) (x + \sqrt{5}) (x^{2} + 4)

(x + \sqrt{5}) (x - \sqrt{5}) (x^{2} - 4)

answer is A.

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Detailed Solution

Given:

x^{4} - x^{2} - 20

= {(x^{2})}^{2} - x^{2} - 20

Let

x^{2} = t

Therefore, after putting the value,

\Rightarrow t^{2} - t - 20 = t^{2} - (5 - 4) t - 20

\Rightarrow t^{2} - t - 20 = t^{2} - 5 {t + 4 t - 20}_{}

\Rightarrow t^{2} - t - 20 = t (t - 5) + 4 (t - 5)

\Rightarrow t^{2} - t - 20 = (t - 5) (t + 4)

Now, substitute the value again,

\Rightarrow {(x^{2})}^{2} - x^{2} - 20 = (x^{2} - 5) (x^{2} + 4)

\Rightarrow x^{4} - x^{2} - 20 = (x^{2} - {(\sqrt{5})}^{2}) (x^{2} + 4)

\Rightarrow x^{4} - x^{2} - 20 = (x + \sqrt{5}) (x - \sqrt{5}) (x^{2} + 4)

[

(a^{2} - b^{2}) = (a + b) (a - b)

]

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