Factoring the Algebraic Expressions

To find the Least Common Multiple (LCM) of the given expressions, we first need to factor each expression completely.

Finding the LCM

The LCM is found by taking the highest power of each unique factor present in the factorizations of the expressions.

The unique factors are:

• \(x^3\) (highest power from Expression 3)
• \(y^2\) (highest power from Expression 3)
• \((x-y)\) (from Expression 3)
• \((x+y)\) (from Expression 2)
• \((x^2 - xy + y^2)\) (present in Expressions 1 and 2, highest power is 1)
• \((x^2 + xy + y^2)\) (from Expression 1)

Multiplying these factors together gives the LCM:

LCM \(= x^3 \cdot y^2 \cdot (x-y) \cdot (x+y) \cdot (x^2 - xy + y^2) \cdot (x^2 + xy + y^2)\)

Simplifying the LCM Expression

We can simplify the product of factors using algebraic identities:

• Recall the difference of cubes: \(a^3 - b^3 = (a-b)(a^2+ab+b^2)\).
• Recall the sum of cubes: \(a^3 + b^3 = (a+b)(a^2-ab+b^2)\).
• Also, recall the difference of sixth powers: \(a^6 - b^6 = (a^3)^2 - (b^3)^2 = (a^3 - b^3)(a^3 + b^3)\).
• Substituting the factored forms of cubes:
• \(x^6 - y^6 = [(x-y)(x^2+xy+y^2)] \cdot [(x+y)(x^2-xy+y^2)]\)
• Rearranging the terms, we see that \((x-y)(x+y)(x^2-xy+y^2)(x^2+xy+y^2)\) is exactly the product of the polynomial factors we found.

Therefore, the LCM can be written as:

LCM \(= x^3 y^2 (x^6 - y^6)\)

Conclusion

By factoring each polynomial and identifying the highest power of all unique factors, we determined the LCM. The LCM of \(x^4+x^2y^2 + y^4\), \(x^3y+y^4\), and \(x^4y^2 - x^3 y^3\) is \(x^3y^2 (x^6 - y^6)\).