UPSC CDS 2 2019 Math Paper

Concept:The LCM of polynomials is the product of the highest powers of all distinct irreducible factors.Explanation:Factor each polynomial completely.$x^{3} + 3x^{2} + 3x + 1 = (x+1)^{3}$.$x^{3} + 5x^{2} + 5x + 4 = (x+4)(x^{2}+x+1)$.$x^{2} + 5x + 4 = (x+4)(x+1)$.List all distinct factors: $(x+1)$, $(x+4)$, and $(x^{2}+x+1)$.For $(x+1)$, the highest power is $3$ (from the first polynomial).For $(x+4)$, the highest power is $1$.For $(x^{2}+x+1)$, the highest power is $1$.Multiply these with their highest exponents to get the LCM.LCM = $(x+1)^{3}(x+4)(x^{2}+x+1)$.Answer:Option A: $(x+1)^{3}(x+4)(x^{2}+x+1)$.