https://file.aiquickdraw.com/mathbot/file/1727387127122cot8t0e4.jpg

The binomial in your image is \((4x - 3y)^5\). To expand this, we will use the **binomial theorem**, which states:

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

Here:
- \(a = 4x\),
- \(b = -3y\),
- \(n = 5\).

The expansion will involve the binomial coefficients \(\binom{5}{k}\) for \(k = 0, 1, 2, 3, 4, 5\), where each term follows the pattern:

\[
\binom{5}{k} (4x)^{5-k} (-3y)^k
\]

Let's calculate the terms one by one:

1. For \(k=0\): \(\binom{5}{0} (4x)^5 (-3y)^0 = 1 \cdot (4x)^5 = 1024x^5\)
2. For \(k=1\): \(\binom{5}{1} (4x)^4 (-3y)^1 = 5 \cdot 256x^4 \cdot (-3y) = -3840x^4y\)
3. For \(k=2\): \(\binom{5}{2} (4x)^3 (-3y)^2 = 10 \cdot 64x^3 \cdot 9y^2 = 5760x^3y^2\)
4. For \(k=3\): \(\binom{5}{3} (4x)^2 (-3y)^3 = 10 \cdot 16x^2 \cdot (-27y^3) = -4320x^2y^3\)
5. For \(k=4\): \(\binom{5}{4} (4x)^1 (-3y)^4 = 5 \cdot 4x \cdot 81y^4 = 1620xy^4\)
6. For \(k=5\): \(\binom{5}{5} (4x)^0 (-3y)^5 = 1 \cdot (-243y^5) = -243y^5\)

So, the expanded form is:

\[
(4x - 3y)^5 = 1024x^5 - 3840x^4y + 5760x^3y^2 - 4320x^2y^3 + 1620xy^4 - 243y^5
\]

Would you like further explanation on any step? 

Here are some related questions to expand on this:
1. How does the binomial theorem apply to higher powers?
2. How do binomial coefficients relate to Pascal's Triangle?
3. What is the general form of a binomial expansion?
4. How does expanding a binomial differ for positive and negative terms?
5. What are some real-world applications of the binomial expansion?

**Tip:** When expanding binomials, ensure the correct application of negative signs to avoid errors, especially when powers of negative numbers are involved.

Learn how to expand the binomial (4x - 3y)^5 using the binomial theorem. This solution includes detailed step-by-step calculations, binomial coefficients, and applying powers of negative numbers. Perfect for students studying algebra and polynomials in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation