Question

The factors of x⁸ + x⁴ + 1 are :

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

(b) (x⁴ + 1 – x²) (x² - 1 + x) (x² + 1 + x)

(c) (x⁴ - 1 + x²) (x² - 1 + x) (x² + 1 + x)

(d) (x⁴ - 1 + x²) (x² + 1 - x) (x² + 1 + x)

Answer 1

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

x⁸ + x⁴ + 1 = x⁸ + 2x⁴ + 1 – x⁴ (Adding and subtracting x⁴)

= (x⁴ + 1)² – (x²)² = (x⁴ + 1 + x²) (x⁴ + 1 – x²) 

= [(x⁴ + 2x² + 1) – x²] (x⁴ + 1 – x²) 

= [(x² + 1)² – (x)²] (x⁴ + 1 – x²) 

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