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Let's address the two parts of the question step by step:

### Part (a) - Derive a formula for the expansion of \((a - b)^3\):

We can use the binomial theorem for this expansion. The binomial theorem states that:

\[
(a - b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} (-b)^k
\]

For \( n = 3 \), the expansion of \( (a - b)^3 \) is:

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

This is the expanded form.

### Part (b) - Write the polynomial \( (2 - 3x)^3 \) using part (a):

Using the formula we derived for \((a - b)^3\), substitute \( a = 2 \) and \( b = 3x \).

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

Now, simplify each term:

\[
= 8 - 3 \cdot 4 \cdot 3x + 3 \cdot 2 \cdot 9x^2 - 27x^3
\]
\[
= 8 - 36x + 54x^2 - 27x^3
\]

Thus, the expanded form of \( (2 - 3x)^3 \) is:

\[
8 - 36x + 54x^2 - 27x^3
\]

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Would you like further details or have any other questions? Here are some related questions for practice:

1. Expand \( (a + b)^3 \) using the binomial theorem.
2. What is the expansion of \( (x - 4)^3 \)?
3. Find the coefficient of \( x^2 \) in the expansion of \( (5 - 2x)^3 \).
4. Expand \( (x - y)^4 \) using the binomial theorem.
5. Derive a formula for the expansion of \( (a + b)^4 \).

**Tip:** Remember that when expanding binomials, the powers of the first term decrease while the powers of the second term increase in each successive term of the expansion.

a) Derive a formula for the expansion of (a - b)^3. b) Write the polynomial (2 - 3x)^3 using part a.

Learn how to derive the binomial expansion of (a - b)^3 and apply it to expand the polynomial (2 - 3x)^3. This problem covers algebraic concepts and step-by-step binomial expansion suitable for high school students.

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