2022 · Q5
What is the HCF of (x⁸-y⁸) and (x⁷-y⁷+x⁵y²-x²y⁵)?
Elementary MathematicsHCF and LCM
- A(x² + y²)
- B(x² - y²)
- C(x³ - y³ - x²y + xy²)
- D(x³ - y³ + x²y - xy²)
Explanation
Factor the second polynomial: x⁷−y⁷+x⁵y²−x²y⁵ = (x²+y²)(x⁵−y⁵) = (x²+y²)(x−y)(x⁴+x³y+x²y²+xy³+y⁴). Also x⁸−y⁸ = (x−y)(x+y)(x²+y²)(x⁴+y⁴). The common part is (x−y)(x²+y²) = x³−y³−x²y+xy², and the leftover co-factors are coprime (at x=2,y=1 they are 17 and 31), so this is the HCF. The keyed option is C.