Question

Factorize the following :
$\longrightarrow \: \boxed{ \bf \: {x}^{4} + \dfrac{2}{x} + \dfrac{1}{ {x}^{6} } }$

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Answer 1

Solution :

To factorise -

x⁴ + 2/x + 1/x⁶

>> 1/x⁶( x¹⁰ + 2x⁵ + 1)

>> 1/x⁶( x¹⁰ + x⁵ + x⁵ + 1)

>> 1/x⁶[ x⁵(x⁵ + 1) + 1(x⁵+1)]

>> 1/x⁶[x⁵+1]²

>> [x⁵+1]²/x⁶

>> [ (x⁵+1)/x³ ]²

>> [ x² + 1/x³]²

This is the required answer .

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Answer 2

Given :-

x⁴ + 2/x + 1/x⁶

To Find :-

Factorize

Solution :-

$\sf x^4+\dfrac{2}{x}+\dfrac{1}{x^6}$

$\sf x^4+\dfrac{2x^5+1}{x^6}$

$\sf \dfrac{x^4\times x^6+2x^5+1}{x^6}$

$\sf\dfrac{x^{(4+6)}+2x^5+1}{x^6}$

$\sf\dfrac{x^{10}+2x^5+1}{x^6}$

Taking out 1/x⁶ common

$\sf\dfrac{1}{x^6}\bigg(x^{10}+2x^5+1\bigg)$

$\sf\dfrac{1}{x^6}\bigg(x^{10}+x^5+x^5+1\bigg)$

$\sf\dfrac{1}{x^6}\bigg(x^5(x^5+1)+1(x^5+1)\bigg)$

$\sf\dfrac{1}{x^6}\bigg((x^5+1)(x^5+1)\bigg)$

$\sf\dfrac{1}{x^6}\bigg((x^5+1)^2\bigg)$

$\sf\dfrac{1}{(x^3)^2}\bigg((x^5+1)^2\bigg)$

$\sf\dfrac{(x^5+1)^2}{(x^3)^2}$

$\sf \bigg(\dfrac{1+x^5}{x^3}\bigg)^2$