Question

Factorise the following:

x⁴ + x² + 1

Answer 1

Given: x⁴ + x² + 1

Step-wise calculation:

1. Consider x⁴ + x² + 1 as a quadratic expression in terms of x², let y = x², then the expression becomes y² + y + 1.

2. We try to factorize this quadratic y² + y + 1 using the form (y + a)(y + b), but the discriminant is Δ = 12 – 4 × 1 × 1 = –3 < 0, so it has no real roots and cannot be factored over the reals.

3. However, the expression x⁴ + x² + 1 can be factorized into the product of two quadratic polynomials with real coefficients as:

x⁴ + x² + 1 = (x² + x + 1)(x² – x + 1)