MHB How can we factor $\displaystyle x^4 + 1$ into the difference of two squares?

[paulmdrdo1]

how to force factor this into the difference of two squares.

$\displaystyle x^4 + 1$

 

[alyafey22]

Re: brushing up on factoring.

$$x^4+1=x^4+2x^2+1-2x^2$$

or

$$x^4+1=x^4-i^2$$

The rest is for you ;)

 

[paulmdrdo1]

Re: brushing up on factoring.

I would get this

$\displaystyle \displaystyle \begin{align*} x^4 + 1 &= x^4 + 2x^2 + 1 - 2x^2 \\& = \left( x^2 + 1 \right) ^2 - \left( \sqrt{2} \, x \right) ^2 \\& = \left( x^2 - \sqrt{2}\, x + 1 \right) \left( x^2 + \sqrt{2}\,x + 1 \right) \end{align*}$

but i want to know what's your reasoning by choosing the term 2x^2?

 

[alyafey22]

Re: brushing up on factoring.

I would get this

$\displaystyle \displaystyle \begin{align*} x^4 + 1 &= x^4 + 2x^2 + 1 - 2x^2 \\& = \left( x^2 + 1 \right) ^2 - \left( \sqrt{2} \, x \right) ^2 \\& = \left( x^2 - \sqrt{2}\, x + 1 \right) \left( x^2 + \sqrt{2}\,x + 1 \right) \end{align*}$

but i want to know what's your reasoning by choosing the term 2x^2?

The idea is complete the square , since if we have for example :

$$x^2+1$$ our first glance suggests converting it to $x^2+2x+1$ which is a complete square hence factorizing will be possible .