UPSC CDS 2 2020 Math Paper

Concept:The HCF of polynomials is the highest-degree polynomial that divides both polynomials completely. Factor each polynomial to find common factors.Explanation:Step 1: Factor $f(x) = x^{6} - 3x^{4} + 3x^{2} - 1$. It matches the pattern $(a - b)^{3} = a^{3} - 3a^{2}b + 3ab^{2} - b^{3}$ with $a = x^{2}$ and $b = 1$. So $f(x) = (x^{2} - 1)^{3}$. Since $x^{2} - 1 = (x - 1)(x + 1)$, we get $f(x) = (x - 1)^{3}(x + 1)^{3}$. Step 2: Factor $g(x) = x^{3} + 3x^{2} + 3x + 1$. It matches the pattern $(a + b)^{3} = a^{3} + 3a^{2}b + 3ab^{2} + b^{3}$ with $a = x$ and $b = 1$. So $g(x) = (x + 1)^{3}$. Step 3: The common factor in both polynomials is $(x + 1)^{3}$. No other common factor appears. Thus, the HCF is $(x + 1)^{3}$.Answer:The HCF of the given polynomials is $(x + 1)^{3}$, which corresponds to option D.