Problem 28 - Algebra and Trigonometry

Question

Find the first three terms in the expansion of $$ \left(x+\frac{1}{x}\right)^{40} $$

Step-by-Step Solution

Answer

The first three terms are: \(x^{40}, 40x^{38}, 780x^{36}\).

Step 1: Identify the Binomial Formula

The expression \(\left(x + \frac{1}{x}\right)^{40}\) can be expanded using the binomial theorem. The binomial theorem states that \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k}b^k\). Here, \(a = x\), \(b = \frac{1}{x}\), and \(n = 40\). We need to find the first three terms of this expansion.

Step 2: Calculate the First Term

The first term in the binomial expansion is given by \(\binom{40}{0} x^{40} \left(\frac{1}{x}\right)^{0}\). This simplifies to \(x^{40}\), since \(\binom{40}{0} = 1\) and any number to the power of 0 is 1.

Step 3: Calculate the Second Term

The second term is \(\binom{40}{1} x^{39} \left(\frac{1}{x}\right)^{1}\). This simplifies to \(40x^{38}\), because \(\binom{40}{1} = 40\) and \(\left(\frac{1}{x}\right)^{1} = \frac{1}{x}\). Multiplying these gives \(40 \cdot x^{39} \cdot \frac{1}{x} = 40x^{38}\).

Step 4: Calculate the Third Term

The third term is \(\binom{40}{2} x^{38} \left(\frac{1}{x}\right)^{2}\). This simplifies to \(780x^{36}\), as \(\binom{40}{2}\) calculates to 780, and \(\left(\frac{1}{x}\right)^2 = \frac{1}{x^2}\). Multiplying these gives \(780 \cdot x^{38} \cdot \frac{1}{x^2} = 780x^{36}\).

Key Concepts

Binomial Expansion

The binomial theorem is a powerful tool in algebra that allows us to expand expressions raised to any power. For a binomial raised to a power \((a+b)^n\), the theorem provides a formula to find each term in the expansion. Each term has a format given by:

• \(\binom{n}{k} a^{n-k} b^k\)

where \(\binom{n}{k}\) is the binomial coefficient. This coefficient is calculated through combinatorics, as we'll discuss later. In our original problem, the expression \(\left(x+\frac{1}{x}\right)^{40}\) is expanded using the same process.
The first three terms of this expansion are calculated using the binomial theorem, starting from \(k = 0\). These terms are important as they illustrate how to apply the theorem step by step.

Combinatorics

Combinatorics is the study of counting, combinations, and permutations and plays a crucial role in the binomial expansion process. The binomial coefficient \(\binom{n}{k}\) is a way to find out how many ways you can choose \(k\) elements from \(n\) elements without considering the order. It is calculated using the formula:

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

The factorial symbol \(!\) denotes a multiplication chain, e.g., \(4! = 4 \times 3 \times 2 \times 1 = 24\). In our example, combinatorics helped us find the coefficients \(\binom{40}{0} = 1\), \(\binom{40}{1} = 40\), and \(\binom{40}{2} = 780\), leading to the first three terms of the binomial expansion.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables (like \(x\)), and arithmetic operations. They are the building blocks of algebra. Understanding these expressions is vital to manipulate and simplify terms, especially when dealing with the binomial theorem.
In our case, the expression \(\left(x+\frac{1}{x}\right)^{40}\) combines both a variable and its reciprocal, which highlights the importance of keeping track of exponents while expanding.
To handle this correctly, one must apply the laws of exponents, such as \(x^a \cdot x^b = x^{a+b}\). In expanding the original expression, each term involves algebraic simplification, requiring careful attention to changes in the exponents with each term's calculation. The first three terms are achieved by simplifying combinations of \(x\) and \(\frac{1}{x}\). This ensures we accurately express each power of \(x\) as required.