Factorise x^4+x^2+1

Question

The question is about factoring an algebraic expression

Answer 1

x^4+x^2+1

assume it factors into (x^2+ax+1)(x^2+bx+1)

multiply the two trinomials together

you get x^4+ax^3+x^2+bx^3+abx^2+bx+x^2+ax+1

simplify...

x^4+(a+b)x^3+(ab+2)x^2+(a+b)x+1

since there is no x^3 term and no x term, a+b must be equal to 0

let a=1 and let b=-1

replace a and b by these numbers in the 5th degree polynomial above and you get the original polynomial

now replace a and b by these numbers in the above factored form, second line,  to get...

(x^2+x+1)(x^2-x+1)

multiply these together to get the original polynomial x^4+x^2+1

you can continue on and use the quadratic equation to factor these two trinomials but you will get complex numbers

Answer 2

The trinomial cannot be factored.