UPSC CDS 2 2023 Math Paper

Concept:Use the relationship between the constant terms of the polynomial and its factors.Explanation:Since $(x-1)^3$ is a factor and the given polynomial is of degree 4, the other factor must be linear.Let the other factor be $(x + k)$.Then the polynomial can be written as $(x-1)^3 \cdot (x + k)$.The constant term of $(x-1)^3$ is $(-1)^3 = -1$.The constant term of $(x + k)$ is $k$.So the constant term of the product is $(-1) \cdot k = -k$.But the constant term of the given polynomial $x^4 + \alpha x^3 + \beta x^2 + \gamma x - 1$ is $-1$.Equate: $-k = -1$, hence $k = 1$.Thus the other factor is $(x + 1)$.Answer:$x + 1$ (Option A).