Problem 35 - Algebra 2

Question

Find the indicated term of each expansion. fifth term of \((2 a+3 b)^{10}\)

Step-by-Step Solution

Answer

The fifth term is \(1088640 a^6 b^4\).

Step 1: Identify the Binomial Theorem

To find a specific term in the expansion of \((a+b)^n\), we use the binomial theorem formula: \[ T_{k+1} = \binom{n}{k} a^{n-k} b^k \]. This formula helps us determine each term in the expansion.

Step 2: Determine Values for the Fifth Term

In this problem, we want the fifth term, which corresponds to \(k = 4\) because terms are counted starting from zero. This means \(n = 10\), \(a = 2a\), \(b = 3b\), and \(k = 4\).

Step 3: Apply Values into Binomial Theorem Formula

Using the values from Step 2, apply them to the formula: \[ T_{5} = \binom{10}{4} (2a)^{10-4} (3b)^4 \].

Step 4: Calculate Each Component

First, calculate the binomial coefficient: \(\binom{10}{4} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210\).Next, calculate \((2a)^{6} = 2^6 a^6 = 64a^6\).Then calculate \((3b)^4 = 3^4 b^4 = 81b^4\).

Step 5: Multiply the Components

Multiply the binomial coefficient with the powers of \(a\) and \(b\): \[ T_{5} = 210 \times 64 a^6 \times 81 b^4 \].Calculate this to get the term: \( T_{5} = 210 \times 64 \times 81 a^6 b^4 = 1088640 a^6 b^4 \).

Key Concepts

Binomial Coefficient

In the context of the binomial theorem, binomial coefficients play a vital role. They tell us how many ways we can choose "k" items from a total of "n" items, and are represented using the symbol \( \binom{n}{k} \). These coefficients are crucial for calculating each term in a binomial expansion.

To find these coefficients, we utilize a simple formula:

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

Here, "!" denotes the factorial operation, meaning you multiply the number by all the positive integers less than itself. For instance, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). In solving problems related to binomial expansion, computing these coefficients helps us understand the contribution of each term. For our specific problem, calculating \( \binom{10}{4} \) gives us 210, which is used in the expansion of the binomial expression.

Polynomial Expansion

Polynomial expansion is a mathematical method used to express a binomial raised to any power \((a + b)^n\) into a sum of terms. Each term in this expansion is affected by a binomial coefficient, the powers of "a", and "b".

The general formula used is:

• \( T_{k+1} = \binom{n}{k} a^{n-k} b^k \)

This formula allows us to determine individual terms in the expansion. To expand \( (2a + 3b)^{10} \), we follow the formula, where each term adheres to this structure based on varying values of "k", the term position.

For the fifth term (keeping in mind that counting starts from zero), applying the formula involves calculating the powers and utilizing the respective binomial coefficient. This structured method ensures that each term is thoroughly calculated, providing the full expanded form of the polynomial expression.

Term Identification

Finding a particular term in a binomial expansion involves understanding the indexing of terms, where terms are labeled starting with the power zero. This means if you need the fifth term, the index "k" is actually 4.

Each term in the binomial expansion \((a+b)^n\) can be represented by:

• \( T_{k+1} = \binom{n}{k} a^{n-k} b^k \)

For our case \((2a + 3b)^{10}\), the task was to find the fifth term. Knowing "n" is 10 and "k" is 4, you plug these values into the formula. Consequently, \( T_5 = \binom{10}{4} (2a)^{6} (3b)^{4} \) gives the specific construction for that fifth term in the expansion.

This method is systematic and ensures accuracy when identifying specific terms in more complex polynomial expansions, guiding you through each necessary step for effective term retrieval. By following these steps, the actual computation results in a term like \( 1088640 a^6 b^4 \).