(2ax - by)3 = Blank 1a3x + Blank 2a2xby + Blank 3axb2y + Blank 4b3y 

To expand the expression \((2ax - by)^3\), we can use the Binomial Theorem, which states:

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

For \((2ax - by)^3\), let’s identify the terms:
- \(a = 2ax\)
- \(b = -by\)
- \(n = 3\)

Now, we can apply the Binomial Theorem:

\[
(2ax - by)^3 = \sum_{k=0}^{3} \binom{3}{k} (2ax)^{3-k} (-by)^k
\]

### Step-by-Step Expansion

1. **Term with \(k = 0\):**
   \[
   \binom{3}{0} (2ax)^3 (-by)^0 = 1 \cdot (2ax)^3 \cdot 1 = 8a^3x^3
   \]

2. **Term with \(k = 1\):**
   \[
   \binom{3}{1} (2ax)^2 (-by)^1 = 3 \cdot (4a^2x^2) \cdot (-by) = -12a^2xbxy
   \]

3. **Term with \(k = 2\):**
   \[
   \binom{3}{2} (2ax)^1 (-by)^2 = 3 \cdot (2ax) \cdot (b^2y^2) = 6ab^2xy^2
   \]

4. **Term with \(k = 3\):**
   \[
   \binom{3}{3} (2ax)^0 (-by)^3 = 1 \cdot 1 \cdot (-b^3y^3) = -b^3y^3
   \]

### Putting It All Together

Combine all the terms:

\[
(2ax - by)^3 = 8a^3x^3 - 12a^2xbxy + 6ab^2xy^2 - b^3y^3
\]

### Numerical Coefficients

Now, let's identify the numerical coefficients for each term:

1. **Blank 1 (for \(a^3x^3\)):** \(8\)
2. **Blank 2 (for \(a^2xbxy\)):** \(-12\)
3. **Blank 3 (for \(ab^2xy^2\)):** \(6\)
4. **Blank 4 (for \(b^3y^3\)):** \(-1\)

Thus, the numerical coefficients are:
- **Blank 1:** 8
- **Blank 2:** \(-12\)
- **Blank 3:** 6
- **Blank 4:** \(-1\)

Would you like more details or have any questions?

### Related Questions:

1. What is the Binomial Theorem, and how is it applied in algebra?
2. Can you explain how to compute binomial coefficients \(\binom{n}{k}\)?
3. How do signs change in binomial expansions involving subtraction?
4. What are some real-world applications of binomial expansions?
5. How would you expand a binomial raised to the fourth power?

**Tip:** When expanding expressions using the Binomial Theorem, carefully handle the signs of the terms, especially if they involve subtraction or negative values.

Learn how to expand the expression (2ax - by)^3 using the Binomial Theorem with a detailed step-by-step solution. This problem covers algebraic expansion techniques, including the application of binomial coefficients. Suitable for students in Grades 10-12.

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