Question

Show that the polynomial f(x) = x⁴ + 4x² + 6 has no zero.

Answer 1

Given: f(x) = x⁴ + 4x² + 6.

Step-wise calculation:

1. Complete the square for the quadratic in x²: 

f(x) = x⁴ + 4x² + 6

= (x² + 2)² + 2

2. For every real x, x² ≥ 0.

Hence, x² + 2 ≥ 2.

So, (x² + 2)² ≥ 4.

3. Therefore, f(x) = (x² + 2)² + 2 ≥ 4 + 2 = 6 > 0 for all real x.

f(x) is always strictly positive for real x, so it has no real zero.