Problem 45 - College Algebra

Question

In the expansion of \((5 x+3 y)^{n},\) each term has the form \(\left(\begin{array}{l}n \\ k\end{array}\right) a^{n-k} b^{k},\) where \(k\) successively takes on the value \(0,1,2, \ldots, n .\) If \(\left(\begin{array}{c}n \\ k\end{array}\right)=\left(\begin{array}{l}7 \\\ 2\end{array}\right),\) what is the corresponding term?

Step-by-Step Solution

Answer

The term is \(590625x^5 y^2\).

Step 1: Identify the Binomial Coefficient

The problem gives us a specific binomial coefficient in the expansion. We know that \( \binom{n}{k} = \binom{7}{2} \). Since \( \binom{n}{k} = \binom{n}{2} \), it implies \( n = 7 \) and \( k = 2 \). This will be important for calculating our term later.

Step 2: Identify Terms a and b

In the binomial expansion of \( (5x + 3y)^n \), we identify \( a = 5x \) and \( b = 3y \). These are the terms that will be raised to powers according to the formula.

Step 3: Apply the General Term Formula

Use the formula for the general term in a binomial expansion: \( T_k = \binom{n}{k} a^{n-k} b^{k} \). Substitute \( n = 7 \), \( k = 2 \), \( a = 5x \), and \( b = 3y \).

Step 4: Calculate the Binomial Coefficient

Calculate \( \binom{7}{2} = \frac{7 \cdot 6}{2 \cdot 1} = 21 \). This will be the coefficient for our term.

Step 5: Calculate Powers of Terms

Calculate \( (5x)^{7-2} = (5x)^5 \) and \( (3y)^2 = 9y^2 \). These calculations involve raising each term to the corresponding power.

Step 6: Combine All Parts to Get the Term

Combine all parts using the formula: \( T_2 = \binom{7}{2} \cdot (5x)^5 \cdot (3y)^2 \). This simplifies to \( 21 \cdot (5^5 x^5) \cdot 9y^2 = 21 \cdot 3125x^5 \cdot 9y^2 = 590625x^5 y^2 \). This is your term corresponding to \( \binom{7}{2} \).

Key Concepts

Binomial Coefficient

The binomial coefficient is an important concept when dealing with binomial expansions. It's used to determine the number of ways to choose \( k \) items from \( n \) items without regard to the order. This is often expressed as \( \binom{n}{k} \).
This formula is crucial for expanding binomials like \((a + b)^n\). The binomial coefficient is calculated using the formula:

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

Where \(!\) denotes a factorial, meaning the product of all positive integers up to a certain number. For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
In our exercise, we used the binomial coefficient \( \binom{7}{2} \), which we calculated to be 21. This number indicates that there are 21 combinations possible when choosing 2 items from a total of 7. Understanding how to compute and apply binomial coefficients is fundamental when solving polynomial expansions.

Polynomial Expansion

Polynomial expansion refers to breaking down expressions like \((a + b)^n\) into their individual terms. The Binomial Theorem provides a formula to expand any binomial raised to a power \( n \).
For a binomial like \( (5x + 3y)^n \), each term in the expansion will be a combination of the two components \( a \) and \( b \), raised to certain powers that add up to \( n \).
The complete expansion for \((a + b)^n\) can be expressed as follows:

• \( \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \)

Here, \( \sum \) means that you sum all the terms together. Each term's coefficients and exponents are determined using the binomial coefficient and the powers of \( a \) and \( b \). In our example, we expanded \( (5x + 3y)^7 \) and computed specific terms based on given coefficients.

General Term Formula

The general term formula in a binomial expansion helps us determine any term in the sequence without needing to expand the entire expression. This is especially useful for large values of \( n \).
The formula for the general term in an expansion \( (a + b)^n \) is given by:

• \( T_k = \binom{n}{k} a^{n-k} b^k \)

Where \( T_k \) represents the \( k \)-th term, \( \binom{n}{k} \) is the binomial coefficient, \( a \) is the first term in the binomial, \( b \) is the second term, and \( k \) is the term’s position starting from 0.
Using this formula, you can identify any term in the series. In our exercise, we identified that \( n = 7 \) and \( k = 2 \), and used the general term formula to compute the precise term in the expansion. The term was calculated as \( 590625x^5 y^2 \), showcasing the practical application of the formula in simplifying polynomial expansions.