Problem 32 - 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 Expression

The given expression is \( \left(x + \frac{1}{x}\right)^{40} \), which represents a binomial expansion problem. We will use the Binomial Theorem to expand this expression.

Step 2: Recall the Binomial Theorem

The Binomial Theorem states that \[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]In this case, \( a = x \) and \( b = \frac{1}{x} \), with \( n = 40 \).

Step 3: Calculate the First Term

The first term \( T_0 \) in the expansion is when \( k=0 \). \[T_0 = \binom{40}{0} x^{40-0} \left(\frac{1}{x}\right)^0 = 1 \cdot x^{40} \cdot 1 = x^{40}\]So, the first term is \( x^{40} \).

Step 4: Calculate the Second Term

The second term \( T_1 \) is when \( k=1 \).\[T_1 = \binom{40}{1} x^{40-1} \left(\frac{1}{x}\right)^1 = 40 \cdot x^{39} \cdot \frac{1}{x} = 40 \cdot x^{38}\]So, the second term is \( 40x^{38} \).

Step 5: Calculate the Third Term

The third term \( T_2 \) is calculated when \( k=2 \).\[T_2 = \binom{40}{2} x^{40-2} \left(\frac{1}{x}\right)^2 = 780 \cdot x^{38} \cdot \frac{1}{x^2} = 780 \cdot x^{36}\]Thus, the third term is \( 780x^{36} \).

Step 6: Write the First Three Terms of the Expansion

Therefore, the first three terms of the expansion of \( \left(x + \frac{1}{x}\right)^{40} \) are:\[ x^{40} + 40x^{38} + 780x^{36} \]

Key Concepts

Binomial Theorem

The Binomial Theorem is a fundamental principle in algebra that provides a way to expand expressions raised to a power, specifically expressions of the form \((a + b)^n\).This theorem states that such expressions can be expanded into a sum involving terms of the form \(\binom{n}{k} a^{n-k} b^k\), where \(\binom{n}{k}\) are the binomial coefficients.

The Binomial Theorem is particularly useful for polynomials and comes in handy for simplifying calculations in larger expansions, such as finding specific terms without expanding the entire expression.Here’s how it works:

• Identify the components of the binomial: \(a\) and \(b\) are the terms being summed, and \(n\) is the power to which the sum is raised.
• Use the sum expression: \(\sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). This is used to determine each term in the expansion.

In our exercise, with \(a = x\), \(b = \frac{1}{x}\), and \(n = 40\), the theorem provides a systematic approach to compute each term in the expansion of \((x + \frac{1}{x})^{40}\).

Binomial Coefficients

Binomial coefficients are the numerical factors that appear in the expanded form of a binomial expression. They are denoted as \(\binom{n}{k}\) and calculated as follows:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]where ! denotes a factorial, meaning the product of all positive integers up to that number.

For example, for \((x + \frac{1}{x})^{40}\), the binomial coefficients help to determine the magnitude of each term in the expansion. The first three coefficients are:

• \(\binom{40}{0} = 1\)
• \(\binom{40}{1} = 40\)
• \(\binom{40}{2} = 780\)

Each coefficient corresponds to a term where the variable powers follow the pattern from the Binomial Theorem. This concept simplifies finding specific terms without full expansion.

Expansion of Powers

The expansion of powers, especially using the binomial theorem, involves expressing powers as a sum of terms. Each term in the expansion has three parts:

• A binomial coefficient, which gives the term's weight.
• A power of the first component of the binomial \((x^{n-k})\).
• A power of the second component \((b^k)\).

In the expansion of \((x + \frac{1}{x})^{40}\), we have:

• The first term \(x^{40}\) results from \(k=0\), meaning there is no \(\frac{1}{x}\) factor.
• The second term \(40x^{38}\), with a coefficient of 40, as \(b^1\) balances the exponents.
• The third term, \(780x^{36}\), becomes more significant as \(b\)'s exponent increases, showing growing symmetrical patterns in expansions.

This method enables efficient calculation of polynomial terms without writing the full polynomial out, saving time and reducing errors in solving problems.