Problem 49 - College Algebra with Modeling and Visualization

Question

Find the specified term. The fourth term of \((2 x+y)^{5}\)

Step-by-Step Solution

Answer

The fourth term is \( 40x^2y^3 \).

Step 1: Understand the problem

We need to find the fourth term in the expansion of \( (2x + y)^5 \) using the binomial theorem.

Step 2: Identify the binomial theorem

The binomial theorem allows us to expand \( (a + b)^n \) as \ \sum_{k=0}^{n} {n \choose k} a^{n-k} b^k \.

Step 3: Determine the general term

The general term in the expansion of \( (2x + y)^{5} \) is \( T_{k+1} = {5 \choose k} (2x)^{5-k} y^k \).

Step 4: Identify the fourth term

Since we're interested in the fourth term \( T_4 \), we need \( T_{3+1} \). Therefore, \( k = 3 \) in our general term formula.

Step 5: Calculate the binomial coefficient

Calculate \( {5 \choose 3} = \frac{5!}{3!(5-3)!} = 10 \).

Step 6: Substitute the values into the general term formula

Substitute \( k = 3 \) into the general term: \[ T_4 = {5 \choose 3} (2x)^{5-3} y^3 \]\[ T_4 = 10 \cdot (2x)^2 \cdot y^3 \]

Step 7: Simplify the expression

Simplify \( (2x)^2 \) to \( 4x^2 \), and substitute back:\[ T_4 = 10 \cdot 4x^2 \cdot y^3 = 40x^2y^3 \]

Step 8: Write the fourth term

The fourth term of the expansion is \( 40x^2y^3 \).

Key Concepts

Binomial Expansion

When we talk about binomial expansion, we are referring to the process of expanding expressions that are raised to a power, such as \( (a+b)^n \), into a sum of terms. This is done using the binomial theorem. The binomial theorem is a powerful tool that provides a formula to expand the given expression in a systematic way. It transforms the expression into a polynomial consisting of terms in the form \( {n \choose k} a^{n-k} b^k \).

Each term is identified by its power or degree which is determined by \( n-k \) for \( a \) and \( k \) for \( b \). The full expansion ends up being a series of these terms from \( k=0 \) all the way up to \( k=n \). This systematic method simplifies the otherwise complex process of performing a multiplication multiple times.

Binomial Coefficient

The binomial coefficient plays a crucial role in the binomial expansion. It is represented as \( {n \choose k} \), which is read as "n choose k." This coefficient is a key component as it tells us how many ways we can choose \( k \) elements from a total of \( n \) elements. This mathematical concept is not only essential in binomial expansion but also in probability and statistics.

To calculate the binomial coefficient, you use the formula \[ {n \choose k} = \frac{n!}{k!(n-k)!} \], where \( n! \) (n factorial) is the product of all positive integers less than or equal to \( n \). For instance, in our problem above, we found that \( {5 \choose 3} = 10 \). These coefficients appear in Pascal's triangle and are symmetric, adding an interesting aspect to their properties.

Algebraic Expressions

Algebraic expressions are a fundamental part of mathematics and include terms that consist of variables, coefficients, and operations such as addition, subtraction, and multiplication. In the context of binomial expansion like \( (2x+y)^5 \), we encounter terms that must be combined and simplified.

For example, after determining the binomial coefficient and substituting the values, the expression may look complex, as we saw: \( 10 \cdot (2x)^2 \cdot y^3 \). It's important to simplify these expressions for a clearer understanding, resulting in \( 40x^2y^3 \). These expressions form the building blocks of equations, helping in solving real-world problems across different fields like physics and engineering.