UPSC CDS 2 2020 Math Paper

Concept:The algebraic identity for the difference of powers states that $(x^n - a^n)$ is always divisible by $(x - a)$ when $n$ is a positive integer.Explanation:We use the factor theorem or polynomial division.For any natural number $n$, the expression $x^n - a^n$ can be factored as $(x - a)(x^{n-1} + x^{n-2}a + \cdots + a^{n-1})$.This is a standard identity that holds for all $n \in \mathbb{N}$ (positive integers).Thus, $(x - a)$ is a factor of $x^n - a^n$ for every natural number $n$, regardless of whether $n$ is even, odd, or prime.The condition $x \neq a$ is given to avoid division by zero, but the divisibility holds algebraically.Therefore, the statement is true for every natural number $n$.Answer:Option A: natural number $n$.