To expand \((x - 6)^3\), you can apply the binomial theorem or use the formula for the cube of a binomial:

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

For \((x - 6)^3\), let \(a = x\) and \(b = 6\). Applying the formula:

\[
(x - 6)^3 = x^3 - 3x^2(6) + 3x(6^2) - 6^3
\]

Now, simplify each term:

\[
x^3 - 18x^2 + 108x - 216
\]

So, the expanded form of \((x - 6)^3\) is:

\[
x^3 - 18x^2 + 108x - 216
\]

Would you like more details or have any questions?

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Here are some related questions you might find helpful:

1. How do you expand \((x + a)^3\) using the binomial theorem?
2. What is the binomial theorem, and how does it apply to binomials raised to powers?
3. Can you expand \((2x - 5)^3\) step by step?
4. How would you factor the expression \(x^3 - 18x^2 + 108x - 216\)?
5. What is the general formula for the expansion of \((a - b)^3\)?
6. How does the sign of \(b\) affect the expansion in \((a \pm b)^3\)?
7. Can you explain the Pascal's triangle and its relation to binomial expansion?
8. How do you determine the roots of a cubic polynomial like \(x^3 - 18x^2 + 108x - 216\)?

**Tip:** When expanding binomials, carefully apply the binomial theorem or the special formulas to avoid sign errors.

Learn how to expand (x - 6)^3 step-by-step using the binomial theorem and understand the formula for the cube of a binomial. This problem involves polynomial expansion and is suitable for students in Grades 10-12.

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