Answer:
The LCM of the three expressions is (a + b) * (a^2 - ab + b^2) * (2a^2 - 2ab + b^2) * (a^2 + b^2)
Explanation:
Expression 1: a^4 + a^2b^2 + b^4
Expression 2: 2a^3 + b^3
Expression 3: a^3 - b^3 + ab^2
Factorizing Expression 1:
a^4 + a^2b^2 + b^4 = (a^2 + b^2)(a^2 + b^2)
Factorizing Expression 2:
2a^3 + b^3 = (a + b)(2a^2 - 2ab + b^2)
Factorizing Expression 3:
a^3 - b^3 + ab^2 = (a + b)(a^2 - ab + b^2)
Now, let's find the LCM by identifying the common factors and the factors unique to each expression:
Expression 1: (a + b)(a^2 + b^2) * (a^2 + b^2)
Expression 2: (a + b)(2a^2 - 2ab + b^2)
Expression 3: (a + b)(a^2 - ab + b^2)
LCM: (a + b) * (a^2 - ab + b^2) * (2a^2 - 2ab + b^2) * (a^2 + b^2)
Therefore, the LCM of the three expressions is:
(a + b) * (a^2 - ab + b^2) * (2a^2 - 2ab + b^2) * (a^2 + b^2)