a² (b² - c²) + b² (c² - a²) + c² (a² - b²)

Let's simplify the expression:

\[
a^2 (b^2 - c^2) + b^2 (c^2 - a^2) + c^2 (a^2 - b^2)
\]

### Step 1: Expand each term

1. **First term**: \( a^2 (b^2 - c^2) \)

\[
a^2 (b^2 - c^2) = a^2 b^2 - a^2 c^2
\]

2. **Second term**: \( b^2 (c^2 - a^2) \)

\[
b^2 (c^2 - a^2) = b^2 c^2 - b^2 a^2
\]

3. **Third term**: \( c^2 (a^2 - b^2) \)

\[
c^2 (a^2 - b^2) = c^2 a^2 - c^2 b^2
\]

### Step 2: Combine all the terms

Now, let's put all the expanded terms together:

\[
a^2 b^2 - a^2 c^2 + b^2 c^2 - b^2 a^2 + c^2 a^2 - c^2 b^2
\]

### Step 3: Simplify the expression

Notice that we can cancel out terms:

- \( a^2 b^2 \) and \( -b^2 a^2 \) cancel out.
- \( -a^2 c^2 \) and \( c^2 a^2 \) cancel out.
- \( b^2 c^2 \) and \( -c^2 b^2 \) cancel out.

Thus, the expression simplifies to:

\[
0
\]

### Final Result:
\[
a^2 (b^2 - c^2) + b^2 (c^2 - a^2) + c^2 (a^2 - b^2) = 0
\]

Would you like further clarification or additional examples? Here are 5 related questions:

1. How can we apply this result in a real-world scenario?
2. What would happen if we used this formula with specific values for \(a\), \(b\), and \(c\)?
3. How do we factor similar expressions for different powers or variables?
4. Can this kind of simplification be generalized to other expressions with different variables?
5. What is the significance of symmetry in algebraic expressions like this?

**Tip**: Always look for symmetry in algebraic expressions, as it can often lead to simplifications and cancellations that make problems easier to solve.

Learn how to simplify the algebraic expression a² (b² - c²) + b² (c² - a²) + c² (a² - b²). This step-by-step guide demonstrates how to expand, combine, and simplify terms, ultimately showing that the expression equals zero. Suitable for students in Grades 8-10.

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