https://file.aiquickdraw.com/mathbot/file/17268459965937l92ur0a.jpg

The expression in the image is \((3x - 2y)^5\).

To expand this, we can use the **Binomial Theorem**, which states:

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

In this case, \(a = 3x\), \(b = -2y\), and \(n = 5\). Applying the binomial expansion formula, we get:

\[
(3x - 2y)^5 = \sum_{k=0}^{5} \binom{5}{k} (3x)^{5-k} (-2y)^k
\]

This will result in 6 terms (since \(n=5\)):

\[
= \binom{5}{0} (3x)^5 (-2y)^0 + \binom{5}{1} (3x)^4 (-2y)^1 + \binom{5}{2} (3x)^3 (-2y)^2 + \dots + \binom{5}{5} (3x)^0 (-2y)^5
\]

Let me compute this expansion explicitly.The binomial expansion of \((3x - 2y)^5\) is:

\[
243x^5 - 810x^4y + 1080x^3y^2 - 720x^2y^3 + 240xy^4 - 32y^5
\]

Would you like a detailed explanation of the steps involved in the binomial expansion?

Here are some related questions you might find helpful:
1. How does the binomial theorem work for negative exponents?
2. Can the binomial theorem be applied to non-integer exponents?
3. How does the binomial theorem relate to Pascal's triangle?
4. What is the role of binomial coefficients in this expansion?
5. How can this expansion be used in practical applications, such as probability?

**Tip:** Always look for symmetry in binomial expansions to verify your results!

Expand the expression (3x - 2y)^5 using the binomial theorem.

Learn how to expand the expression (3x - 2y)^5 using the binomial theorem. This solution covers binomial coefficients, exponents, and algebraic manipulation. Suitable for students in Grades 10-12.

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