The expression \((a - b)^3\) can be expanded using the binomial theorem or by multiplying it step by step. Here's the expansion:

\[
(a - b)^3 = (a - b)(a - b)(a - b)
\]

Now, applying the distributive property (or FOIL method for the first two terms) step by step:

1. First, expand \((a - b)(a - b)\):

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

2. Then, multiply the result by \((a - b)\):

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

Now, distribute each term in \((a^2 - 2ab + b^2)\) by \((a - b)\):

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

Expanding each term:

\[
= a^3 - a^2b - 2a^2b + 2ab^2 + ab^2 - b^3
\]

Combining like terms:

\[
= a^3 - 3a^2b + 3ab^2 - b^3
\]

Therefore, the expanded form of \((a - b)^3\) is:

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

Would you like to go over the steps in more detail or have any questions? Here are some related questions:

1. What is the expanded form of \((a + b)^3\)?
2. How can we expand \((a - b)^4\)?
3. How does the binomial theorem apply to higher powers of binomials?
4. What are the coefficients in the expansion of \((a - b)^n\) for various \(n\)?
5. How can we factor the expression \(a^3 - 3a^2b + 3ab^2 - b^3\)?

**Tip:** When expanding binomials, remember the pattern of coefficients follows Pascal's Triangle, which can help simplify calculations for higher powers!

Discover the step-by-step expansion of the binomial expression (a - b)^3 using the binomial theorem and distributive property. This problem demonstrates key algebraic principles, such as polynomial expansion and simplifying like terms, suitable for students in Grades 9-11.

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