https://file.aiquickdraw.com/mathbot/file/17267151117519e20zeq7.jpg

$$ x^2+ $$$$ xy+y^3\text{aur }x^3-y^3 $$$$ ka\text{ LCM} $$

The new image you uploaded contains the following mathematical expression:

\[
x^2 + xy + y^3 \quad और \quad x^3 - y^3 \quad का \quad LCM \quad निकालिए
\]

This translates to:

"Find the LCM (Least Common Multiple) of $x^2 + xy + y^3$ and $x^3 - y^3$."

### Solution:

1. **Expression 1:**
   \[
   x^2 + xy + y^3
   \]
   This expression does not follow any standard factorization rules, so we leave it as it is.

2. **Expression 2:**
   \[
   x^3 - y^3
   \]
   This is a difference of cubes, and it can be factorized as:
   \[
   x^3 - y^3 = (x - y)(x^2 + xy + y^2)
   \]

### Finding the LCM:

- The first expression $x^2 + xy + y^3$ does not share a direct factorization with the second expression.
- The second expression $x^3 - y^3$ has the factorization $(x - y)(x^2 + xy + y^2)$, but this does not directly include $y^3$, so no common factor exists.

The LCM will be the product of the two expressions, as they do not share a common factor:
\[
LCM = (x^2 + xy + y^3)(x - y)(x^2 + xy + y^2)
\]

Thus, the LCM is:
\[
LCM = (x^2 + xy + y^3)(x - y)(x^2 + xy + y^2)
\]

Would you like more details or any clarification?

### Related Questions:
1. How do we find the LCM when no common factors exist?
2. What is the factorization of higher-degree polynomials?
3. Can you explain how the difference of cubes formula is derived?
4. How can we simplify complex expressions like the one above?
5. What are the key differences between polynomial LCM and number LCM?

**Tip:** When dealing with polynomial LCM, always check if the terms share a common factor before multiplying the entire expressions.

Find the LCM of the expressions x^2 + xy + y^3 and x^3 - y^3.

Learn how to find the Least Common Multiple (LCM) of the polynomials x^2 + xy + y^3 and x^3 - y^3. This solution explains the application of the difference of cubes formula and polynomials' LCM computation, suitable for students in Grades 10-12.

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