2026 · Q12
Consider the polynomial p(x) = x⁵+2x⁴+x³−x²−2x−1 :
- Ix²+2x+1 is a factor of p(x).
- IIx²+x+1 divides p(x).
Which of the statements given above is/are correct?
Elementary MathematicsAlgebra
- AI only
- BII only
- CBoth I and II
- DNeither I nor II
Explanation
For statement I, p(−1) = −1 + 2 − 1 − 1 + 2 − 1 = 0 and p'(x) = 5x⁴ + 8x³ + 3x² − 2x − 2 gives p'(−1) = 5 − 8 + 3 + 2 − 2 = 0, so (x + 1)² = x² + 2x + 1 is a factor — statement I is correct. For statement II, with ω a non-real cube root of unity (ω³ = 1, ω² + ω + 1 = 0), p(ω) = ω² + 2ω + 1 − ω² − 2ω − 1 = 0 and likewise p(ω²) = 0, so x² + x + 1 divides p(x) — statement II is correct. Hence option C.