​What is the HCF of x⁴- 13x²y²- 300y⁴, x³ - 4x²y - 4xy² - 5y³and x³- 125y³?​

Correct option is A

Given:

x⁴ - 13x²y² - 300y⁴,

x³- 4x²y - 4xy² - 5y³,

x³- 125y³

Solution:    

x³ - 125y³= (x)² - (5y)² = (x - 5y)(x² + 25y² + 5xy) ..... (i)

x⁴ - 13x²y² - 300y⁴ = x⁴ - 25x²y² + 12x²y² - 300y²

x²(x² - 25y²) + 12y²(x² - 25y²)

(x² + 12y²)(x² - 25y²) 

(x² + 12y²)(x - 5y)(x + 5y) .....(ii)

Since (x - 5y) is zero of both the polynomial (i) and (ii)

Put x - 5y = 0, x = 5y in third polynomial

to confirm (x - 5y) as HCF,

x³ - 4x²y - 4xy²- 5y³ = 125y³ - 100y³ -20y³ -5y³ = 0

Hence HCF is (x - 5y).

∴ The correct answer is x - 5y.