Question

factorise x⁴+1/x⁴+1​

Answer 1

Answer:

factorise x⁴+1/x⁴+1

2

Answer 2

$\huge\star{\underline{\underline{\sf{ Solution:-}}}}$

$\implies{\sf{ \left( x^4 + \dfrac{1}{x^4} + 1 \right) }}$

$\implies{\sf{ \left( x^4 + \dfrac{1}{x^4} + 2 \right) - 1 = \left( x^2 + \dfrac{1}{x^2} \right)^2 - 1^2 }}$

$\implies{\sf{ \left( x^2 + \dfrac{1}{x^2} - 1 \right) \left(x^2 + \dfrac{1}{x^2} + 1 \right)}}$

$\implies{\sf{ \left( x^2 + \dfrac{1}{x^2} - 1 \right) \left{ \left(x^2 + \dfrac{1}{x^2} + 2 \right) - 1 \right} }}$

$\implies{\sf{ \left( x^2 + \dfrac{1}{x^2} - 1 \right) \left{ \left(x + \dfrac{1}{x} \right)^2 - 1^2 \right} }}$

$\implies{\sf{ \left( x^2 + \dfrac{1}{x^2} - 1 \right) \left(x + \dfrac{1}{x} - 1 \right) \left(x+ \dfrac{1}{x} +1 \right) }}$

Thus,

$\implies\red{\sf{ \left( x^4 + \dfrac{1}{x^4} + 1 \right) = \left( x^2 + \dfrac{1}{x^2} - 1 \right) \left(x + \dfrac{1}{x} - 1 \right) \left(x+ \dfrac{1}{x} +1 \right) }}.$

Extra Dose:

Related Identities:

• (a²-b²) = (a-b) (a+b)
• (a+b)² = a² + b² + 2ab
• ( a -b )² = a² + b² - 2ab