CDS 2025 · PYQ · Polynomials / HCF · medium
What is the HCF of the polynomials x⁸ + x⁴ + 1 and x⁴ + x² + 1?
Answer
The correct answer is C: x⁴ + x² + 1. x⁴ + x² + 1 = (x² + x + 1)(x² − x + 1). Also x⁸ + x⁴ + 1 = (x⁴ + x² + 1)(x⁴ − x² + 1).
- A.1
- B.x⁴ − x² + 1
- C.x⁴ + x² + 1✓ Correct
- D.x⁴ − x² − 1
x⁴ + x² + 1 = (x² + x + 1)(x² − x + 1). Also x⁸ + x⁴ + 1 = (x⁴ + x² + 1)(x⁴ − x² + 1). So x⁴ + x² + 1 divides x⁸ + x⁴ + 1 completely. Hence HCF = x⁴ + x² + 1.
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