Resolve in factors. I) x⁴+x²+1

let a = x² (a ≥ 0), we have the equation:

a² + a + 1 = 0

⇔ a² + 2.1/2.a + 1/4 + 3/4 = 0

⇔ (a + 1/2)² = -3/4 (unreasonable)

=> no solutions

Step-by-step explanation:

(x2−x+1)(x2+x+1)

Step-by-step explanation:

Let’s multiply by x2−1

(x4+x2+1)(x2–1)=x6+x4+x2−x4−x2−1=x6−1

So x4+x2+1=x6−1x2−1 .

Now, let’s factorize x6−1 differently: x6−1=(x3+1)(x3−1) . Also x2−1=(x+1)(x−1) .

So x4+x2+1=(x3+1)(x3−1)(x+1)(x−1)=x3+1x+1⋅x3−1x−1 .

Now, I can factorize x3+1=(x+1)(x2−x+1) and x3−1=(x−1)(x2+x+1) .

So. x4+x2+1=(x2−x+1)(x2+x+1) .