Problem 31 - College Algebra with Corequisite Support

Question

For the following exercises, find the indicated term of each binomial without fully expanding the binomial. The fourth term of \((3 x-2 y)^{5}\)

Step-by-Step Solution

Answer

The fourth term is \(-720x^2y^3\).

Step 1: Understand the Binomial Theorem

The Binomial Theorem describes the expansion of a power of a binomial, \((a+b)^n\), as a sum of terms of the form \( \binom{n}{k} a^{n-k} b^k\). Here, \(a = 3x\), \(b = -2y\), and \(n = 5\).

Step 2: Identify the Position of the Term

To find the fourth term, recognize that terms in the binomial expansion are indexed starting from zero. Therefore, the fourth term corresponds to \(k = 3\).

Step 3: Apply the Binomial Coefficient

Use the formula of the binomial coefficient for the term: \( T_{k+1} = \binom{n}{k} a^{n-k} b^k \). For this case, \(T_4 = \binom{5}{3} (3x)^{5-3} (-2y)^3\). Calculate \(\binom{5}{3} = 10\).

Step 4: Calculate Powers of Each Term

Now compute the powers: \((3x)^{2} = 9x^2\) and \((-2y)^3 = -8y^3\).

Step 5: Put It All Together

Substitute these values into the term formula: \( T_4 = 10 \times 9x^2 \times -8y^3\). Multiply these together to get \( T_4 = -720x^2y^3\).

Key Concepts

Binomial Coefficient

The Binomial Coefficient is a key concept in combinatorics and helps us determine how many ways subsets of items can be selected from a larger set. It is often represented as \( \binom{n}{k} \), which reads as "n choose k". This represents the number of ways to choose \(k\) elements from a set of \(n\) elements without regard to order. In mathematical terms, it's calculated as:

• \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \)

The exclamation point (!) denotes factorial, which is the product of an integer and all the integers below it. For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).
In the context of the Binomial Theorem, the binomial coefficient determines the number of terms or contributions each part of the expanded binomial expression gives. For instance, in the example, \( \binom{5}{3} \) equates to 10, indicating how many ways you can select 3 terms from a set of 5.

Binomial Expansion

Binomial Expansion refers to expressing a binomial raised to a power in the form of a sum. According to the Binomial Theorem, any binomial raised to an exponent can be expanded into a series of terms. For the binomial \((a+b)^n\), the expansion involves terms of the form \( \binom{n}{k} a^{n-k} b^k \).
In practical terms, this means that instead of multiplying the binomial by itself multiple times, we use the theorem to systematically generate each term in the expanded polynomial. Each term involves:

• A binomial coefficient \( \binom{n}{k} \).
• A power of the first term in the binomial \(a^{n-k}\).
• A power of the second term in the binomial \(b^k\).

In the exercise at hand, \((3x - 2y)^5\) is expanded using the theorem, with specific focus on finding the fourth term directly using the formula.

Exponentiation

Exponentiation is a mathematical operation involving numbers or variables raised to the power of another number. It is represented as \(x^n\), where \(x\) is the base and \(n\) is the exponent. The operation involves multiplying the base by itself as many times as the value of the exponent indicates.
In the context of binomial expansions, exponentiation is used within each term to manage the powers of both components of the binomial. For example:

• \((3x)^2\) is computed as \(3x \times 3x = 9x^2\).
• \((-2y)^3\) is calculated as \((-2y) \times (-2y) \times (-2y) = -8y^3\).

Effects of exponentiation are significant in determining the individual contributions of \(a\) and \(b\) in terms within the binomial expansion.