What is the LCM of x^4 + x^2y^2 + y^4, x^3y + y^4 and x^4 y^2 - x^3y^3?
- A. x^3 y^3 (x^6 - y^6)
- B. x^3y^2 (x^6 - y^6) ✓
- C. x^3y(x^6 - y^6)
- D. xy(x^6 - y^6)
Correct Answer: B. x^3y^2 (x^6 - y^6)
Explanation
Factorizing: 1) x^4+x^2y^2+y^4 = (x^2-xy+y^2)(x^2+xy+y^2). 2) x^3y+y^4 = y(x^3+y^3) = y(x+y)(x^2-xy+y^2). 3) x^4y^2-x^3y^3 = x^3y^2(x-y). The LCM collects the highest powers of all factors: x^3 \cdot y^2 \cdot (x-y)(x+y)(x^2-xy+y^2)(x^2+xy+y^2) = x^3y^2(x^2-y^2)(x^4+x^2y^2+y^4) = x^3y^2(x^6-y^6).
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