https://r2.erweima.ai/mathbot/image/17244964441934lqngp12.jpg

To analyze the image, let's transcribe the mathematical expression shown:

The expression is:
\[
a^3 + b^3 + c^3 - 3abc = (a+b+c)(a^2 + b^2 + c^2 - ab - ac - bc)
\]

We need to prove this identity.

### Proof:

Start with the left-hand side (LHS) of the equation:
\[
a^3 + b^3 + c^3 - 3abc
\]

This expression is known as the factorization of the sum of cubes minus three times the product of \(a\), \(b\), and \(c\). It can be factored as:
\[
a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - ac - bc)
\]

**Proof by expansion:**

1. **Expand the right-hand side (RHS):**
   \[
   (a+b+c)(a^2 + b^2 + c^2 - ab - ac - bc)
   \]

2. **Distribute each term in \((a + b + c)\):**
   \[
   a(a^2 + b^2 + c^2 - ab - ac - bc) + b(a^2 + b^2 + c^2 - ab - ac - bc) + c(a^2 + b^2 + c^2 - ab - ac - bc)
   \]

3. **Expand each term:**
   \[
   a^3 + ab^2 + ac^2 - a^2b - a^2c - abc + b^3 + ba^2 + bc^2 - b^2a - b^2c - abc + c^3 + ca^2 + cb^2 - c^2a - c^2b - abc
   \]

4. **Combine like terms:**
   \[
   a^3 + b^3 + c^3 + (ab^2 + ba^2) + (ac^2 + ca^2) + (bc^2 + cb^2) - (a^2b + b^2a) - (a^2c + c^2a) - (b^2c + c^2b) - 3abc
   \]

5. **Notice that the cross terms cancel:**
   \[
   (ab^2 + ba^2 - a^2b - b^2a) = 0
   \]
   \[
   (ac^2 + ca^2 - a^2c - c^2a) = 0
   \]
   \[
   (bc^2 + cb^2 - b^2c - c^2b) = 0
   \]

Thus, we are left with:
\[
a^3 + b^3 + c^3 - 3abc
\]

This matches the left-hand side (LHS), proving the identity.

Would you like more details on this proof or have any questions? Here are some related questions to explore further:

1. How can we factor other polynomial expressions?
2. What are the conditions for this factorization to be valid?
3. Are there other similar identities involving the sums of powers of variables?
4. How does this identity relate to symmetric polynomials?
5. Can this proof be generalized to more variables or higher powers?

**Tip:** When proving identities, always consider both expansion and factorization to see if terms can be simplified or matched between sides of the equation.

Learn how to prove the sum of cubes formula a^3 + b^3 + c^3 - 3abc step-by-step with detailed mathematical expansion and factorization. This problem covers key algebraic concepts including factorization and polynomial identities, suitable for advanced high school students.

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