2023 · Q43
Consider the following statements in respect of the polynomial 1 - x - xⁿ + xⁿ⁺¹ where n is a natural number :
- 1It is divisible by 1 - 2x + x².2. It is divisible by 1 - xⁿ.
Which of the statements given above is/are correct?
Elementary MathematicsAlgebra
- A1 only
- B2 only
- CBoth 1 and 2
- DNeither 1 nor 2
Explanation
Factorising: 1 − x − xⁿ + xⁿ⁺¹ = (1 − x)(1 − xⁿ), so the polynomial is divisible by 1 − xⁿ (statement 2). It is also divisible by (1 − x)² = 1 − 2x + x² (statement 1), since (1 − xⁿ)/(1 − x) = 1 + x + … + xⁿ⁻¹ is a polynomial. Both statements are correct (C).