We expanded \((a+b)^3\) in the initial step to recognize its structure and see how it fits with the formula for the difference of cubes. Here's a more detailed explanation of why we expand it:

### Purpose of Expansion

1. **Understanding the Polynomial:** By expanding \((a+b)^3\), we break it down into a sum of terms involving \(a\) and \(b\). This helps us understand the individual components of the polynomial.
   
2. **Using Algebraic Identities:** The expansion helps us identify patterns or structures that can be used in conjunction with other algebraic identities. In this case, recognizing it as part of the difference of cubes formula.
   
3. **Simplification:** Expansion allows us to combine and simplify expressions more efficiently, leading to a more straightforward final result.

### Alternative Approach Without Expansion

However, if we directly recognize \((a+b)^3 - 8c^3\) as a difference of cubes, we can apply the formula for the difference of cubes directly:

Given:
\[
(a+b)^3 - 8c^3
\]

Notice that \(8c^3 = (2c)^3\). So we have a difference of cubes:
\[
(a+b)^3 - (2c)^3
\]

Using the difference of cubes formula:
\[
x^3 - y^3 = (x - y)(x^2 + xy + y^2)
\]

Where \(x = a+b\) and \(y = 2c\):

\[
(a+b)^3 - (2c)^3 = ((a+b) - 2c)((a+b)^2 + (a+b)(2c) + (2c)^2)
\]

So, directly applying the formula gives us:

\[
(a+b)^3 - 8c^3 = (a+b - 2c)(a^2 + 2ab + b^2 + 2ac + 2bc + 4c^2)
\]

Thus, expanding \((a+b)^3\) initially helps us to see the components, but it's not strictly necessary if we directly recognize and apply the difference of cubes formula.

Would you like more details or have any questions?

Here are some questions you might want to ask next:
1. How do I apply the binomial theorem to higher powers?
2. Can you explain the steps of simplifying a different polynomial expression?
3. How do you factor trinomials?
4. What are some examples of using the difference of squares formula?
5. How do you solve polynomial equations?

**Tip:** Recognizing patterns and knowing when to apply specific algebraic identities can save time and simplify your calculations.

Explore the reasons behind expanding (a+b)³ in algebra, including its significance in recognizing polynomial structure and utilizing the difference of cubes formula. Perfect for students in Grades 8-10.

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