Question

factorize x^4+x^2+1 plz

Answer 1

Factor the following as a product of two quadratics:
x^4 + x^2 + 1
Set x^4 + x^2 + 1 equal to the desired form (x^2 + a x + b) (x^2 + c x + d):
x^4 + x^2 + 1 = (x^2 + a x + b) (x^2 + c x + d)
(x^2 + a x + b) (x^2 + c x + d) = x^4 + (a + c) x^3 + (b + a c + d) x^2 + (b c + a d) x + b d:
x^4 + x^2 + 1 = x^4 + x^3 (a + c) + x^2 (b + a c + d) + x (b c + a d) + b d
The coefficients of x^3 are a + c and 0.
The coefficients of x^2 are b + a c + d and 1.
The coefficients of x are b c + a d and 0.
The constant terms are b d and 1:
{a + c = 0
b + a c + d = 1
b c + a d = 0
b d = 1
The system gives the solution {a = -1
b = 1
c = 1
d = 1:
a = -1 and b = 1 and c = 1 and d = 1
(x^2 + a x + b) (x^2 + c x + d) = (x^2 - x + 1) (x^2 + x + 1):
Answer: |
| (x^2 - x + 1) (x^2 + x + 1)