Problem 9 - Discrete Mathematics with Applications

Question

Find the middle term in the binomial expansion of each. $$\left(x+\frac{1}{x}\right)^{4}$$

Step-by-Step Solution

Answer

The middle term of the binomial expansion of \(\left(x+\frac{1}{x}\right)^{4}\) is 6.

Step 1: Identify a, b, and n.

In this problem, \(a = x\), \(b = \frac{1}{x}\), and \(n = 4\).

Step 2: Find the middle term using the binomial theorem.

Since we have four terms in our expansion, the middle term is the third term. Therefore, we need to find the \(r = 3\) term. According to the binomial theorem, this term is: \(\binom{n}{r-1}a^{n-r+1}b^{r-1} = \binom{4}{3-1}x^{4-3+1}\left(\frac{1}{x}\right)^{3-1}\)

Step 3: Compute the binomial coefficient.

Using the formula for the binomial coefficient, we can compute \(\binom{4}{2}\) as follows: \(\binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4!}{2!2!} = \frac{4 \times 3 \times 2 \times 1}{(2\times 1)(2 \times 1)} = \frac{24}{4} = 6\)

Step 4: Substitute the binomial coefficient and simplify the term.

Now that we have found the binomial coefficient, we can substitute it into our expression and simplify the term: \(6x^{2}\left(\frac{1}{x}\right)^2 = 6x^{2}\frac{1}{x^2} = 6\frac{x^2}{x^2} = 6\)

Step 5: Write the final answer.

The middle term of the binomial expansion of \(\left(x+\frac{1}{x}\right)^{4}\) is 6.

Key Concepts

Binomial Coefficient

The binomial coefficient is an essential part of the binomial theorem and is represented as \( \binom{n}{r} \), which is read as "\( n \) choose \( r \)." This number determines how many ways you can combine \( n \) items taken \( r \) at a time, regardless of order. It is given by the formula:
\[\binom{n}{r} = \frac{n!}{r!(n-r)!}\]
where \( n! \) ("\( n \) factorial") is the product of all positive integers up to \( n \), and \( r! \) is the product of all positive integers up to \( r \).
Quick Points about Binomial Coefficients:

• The coefficient is always a whole number.
• It can be visualized using Pascal's Triangle, where each number is the sum of the two numbers directly above it.
• When \( r = 0 \) or \( r = n \), \( \binom{n}{r} = 1 \).

Understanding binomial coefficients is essential because they give us the necessary multipliers for each term in a binomial expansion. In the exercise, the binomial coefficient \( \binom{4}{2} = 6 \) was used to determine the multiplier for the middle term.

Binomial Theorem

The Binomial Theorem provides a powerful way to expand expressions of the form \( (a + b)^n \). According to the theorem, the expansion is expressed as a sum of terms of the form:\[\binom{n}{r} a^{n-r} b^r\]This concise expression tells us to combine the coefficients (binomial coefficients), the powers of \( a \), and the powers of \( b \), creating every possible product. Each term in the expansion will have a unique power of \( a \) and \( b \) that adds up to \( n \).
Important Features of the Binomial Theorem:

• There are \( n + 1 \) terms in the expansion of \( (a + b)^n \).
• The powers of \( a \) decrease from \( n \) to 0, while the powers of \( b \) increase from 0 to \( n \).
• The coefficients for the terms can be found using binomial coefficients.

In our exercise, the theorem was used to find the third term in the binomial expansion by plugging the specific values of \( a = x \), \( b = \frac{1}{x} \), and \( n = 4 \) into the equation for the binomial expansion.

Middle Term in Expansion

Finding the middle term in a binomial expansion can be straightforward when \( n \), the exponent in \( (a + b)^n \), is even. The middle term(s) is generally found by looking at term \( T \), where \( T_{r+1} \) corresponds to the number of terms divided by 2 plus one for expansions with an odd number of terms or just \( n/2 \) for even \( n \).
For example:

• When \( n = 4 \) (as in our exercise), the total number of terms is 5, so the middle term is the third one, \( T_3 \).
• This term can be calculated using the formula from the binomial theorem discussed earlier: \( \binom{4}{2} x^{2} \left(\frac{1}{x}\right)^2 \).

By simplifying the given expression, you can determine that the middle term in the expansion is 6. Understanding how to find it helps you manage complex polynomial expressions without expanding the entire binomial expression, especially for large \( n \). It showcases the significance of focusing on specific coefficients and their powers.