Problem 29 - Algebra 2

Question

Expand each power. $$ (3 x-2 y)^{5} $$

Step-by-Step Solution

Answer

The expanded form is \(243x^5 - 810x^4y + 1080x^3y^2 - 720x^2y^3 + 240xy^4 - 32y^5\).

Step 1: Recognize the Binomial

The expression \((3x - 2y)^5\) is a binomial raised to the fifth power. To expand this, we'll use the Binomial Theorem.

Step 2: Apply the Binomial Theorem

The Binomial Theorem provides us the formula \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\) to expand \((3x - 2y)^5\). Here, \(a = 3x\), \(b = -2y\), and \(n = 5\).

Step 3: Calculate Each Term of the Expansion

Compute the terms of the expansion using \(\binom{5}{k}\), \((3x)^{5-k}\), and \((-2y)^k\) for \(k = 0\) to \(5\).

Step 4: Term Calculations

- For \(k=0\), the term is \(\binom{5}{0}(3x)^5(-2y)^0 = 1 \, \cdot \, 243x^5 \, \cdot \, 1 = 243x^5\).- For \(k=1\), the term is \(\binom{5}{1}(3x)^4(-2y)^1 = 5 \, \cdot \, 81x^4 \, \cdot \, -2y = -810x^4y\).- For \(k=2\), the term is \(\binom{5}{2}(3x)^3(-2y)^2 = 10 \, \cdot \, 27x^3 \, \cdot \, 4y^2 = 1080x^3y^2\).- For \(k=3\), the term is \(\binom{5}{3}(3x)^2(-2y)^3 = 10 \, \cdot \, 9x^2 \, \cdot \, -8y^3 = -720x^2y^3\).- For \(k=4\), the term is \(\binom{5}{4}(3x)^1(-2y)^4 = 5 \, \cdot \, 3x \, \cdot \, 16y^4 = 240xy^4\).- For \(k=5\), the term is \(\binom{5}{5}(3x)^0(-2y)^5 = 1 \, \cdot \, 1 \, \cdot \, -32y^5 = -32y^5\).

Step 5: Combine the Terms

Add the calculated terms together: \[243x^5 - 810x^4y + 1080x^3y^2 - 720x^2y^3 + 240xy^4 - 32y^5\]

Key Concepts

Polynomial Expansion

The process of polynomial expansion involves expressing a power of a binomial, such as \((3x - 2y)^5\), as a sum of individual terms. This is achieved by using the Binomial Theorem, which simplifies the expansion of expressions raised to powers without manually multiplying the binomial.
Consider the binomial \((a + b)^n\). To expand it, you will write it as a series or sum of several terms. Each term in this expansion is found by following a specific pattern, which is defined by the binomial coefficients and the exponents assigned to each part of the binomial. Here’s a structured method:

• Apply the formula: \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\)
• Compute each term individually using the value of \(k\)
• Sum all the terms together to achieve the final expanded polynomial

Utilizing this technique allows mathematicians and students to expand binomials efficiently without error.

Binomial Coefficients

When expanding a binomial, the coefficients of each term are called binomial coefficients. They describe how many ways you can choose a certain number of items from a larger set. In math terms, this is represented by \(\binom{n}{k}\), which denotes a combination of \(n\) items taken \(k\) at a time.
For example, when dealing with \((3x - 2y)^5\), we calculate the binomial coefficients for \(n=5\) and various values of \(k\), ranging from 0 to 5. You obtain coefficients like 1, 5, and so on.
Here's how you calculate them:

• \(\binom{5}{0} = 1\)
• \(\binom{5}{1} = 5\)
• \(\binom{5}{2} = 10\)
• \(\binom{5}{3} = 10\)
• \(\binom{5}{4} = 5\)
• \(\binom{5}{5} = 1\)

These numbers not only appear in the coefficients of the expanded polynomial but are also found in Pascal's Triangle, providing a fascinating link between combinatorics and algebra.

Pascal's Triangle

Pascal's Triangle is a convenient, triangular arrangement of numbers that provides the binomial coefficients for any binomial expansion. To visualize it easily, each number is the sum of the two directly above it. This pattern continues infinitely. For the binomial \((3x - 2y)^5\), you would look at the sixth row of Pascal’s Triangle (remember it starts with zero index from top), which is where you find the coefficients 1, 5, 10, 10, 5, 1.
Pascal's Triangle is not only great for finding binomial coefficients—it also has fascinating mathematical properties:

• Symmetry: Each row of Pascal’s Triangle is symmetric.
• Binary Representations: Represents the number of combinations, reflecting in each row.
• Links to Powers: Helps in quickly determining powers of numbers like 11.

This simple and intriguing tool highlights the connection between different arithmetic concepts and serves as a quick reference to find the necessary binomial coefficients for polynomial expansions.