quick polynomial question (1 Viewer)

[kooltrainer]

how do u factorise x^4 + x^2 + 1 over rational number system?
it becomes (x^2 + x +1) ( x^2 - x +1 ) and i don't get how because the factor theorm doesnt work.

 

[Trebla]

x⁴ + x² + 1
= x⁴ + 2x² + 1 - x²
= (x² + 1)² - x²
= (x² + 1 - x)(x² + 1 + x)

 

[vafa]

An alternative Solution (By the help of geometric series):

View attachment 15829

 

[vafa]

by the way, I think it is year 10 work.

 

[kooltrainer]

It is trivial:

x⁴ + x² + 1 is quadratic in form, i.e. it is the same as

u² + u + 1 where u = x², since x⁴ ≡ (x²)².

Therefore you can factorise it as you would a quadratic simply by substituting, though it is recommended that you don't substitute explicitly but in your head since it involves so little working.

how would that help u factorise .. thats only solving x and is needed for factorising over complex..