Problem 32 - Precalculus Mathematics for Calculus

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: Understanding the Binomial Theorem

The expression \((x+\frac{1}{x})^{40}\) is expanded using the binomial theorem. According to this theorem, \\[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]For \(\left(x+\frac{1}{x}\right)^{40}\), let \(a = x\), \(b = \frac{1}{x}\), and \(n = 40\).

Step 2: Finding the First Term

For the first term, use \(k = 0\):\[\binom{40}{0} x^{40-0} \left(\frac{1}{x}\right)^0 = 1 \cdot x^{40} = x^{40}\]Thus, the first term is \(x^{40}\).

Step 3: Finding the Second Term

For the second term, use \(k = 1\):\[\binom{40}{1} x^{40-1} \left(\frac{1}{x}\right)^1 = 40 \cdot x^{39} \cdot \frac{1}{x} \]Simplifying, we get \(40x^{38}\). Thus, the second term is \(40x^{38}\).

Step 4: Finding the Third Term

For the third term, use \(k = 2\):\[\binom{40}{2} x^{40-2} \left(\frac{1}{x}\right)^2 \]First, calculate \(\binom{40}{2} = \frac{40 \cdot 39}{2} = 780\). Then: \780 \cdot x^{38} \cdot \frac{1}{x^2} = 780x^{36}\Thus, the third term is \(780x^{36}\).

Key Concepts

Binomial Expansion

The binomial expansion is a powerful tool in algebra, especially when expanding expressions raised to large powers. This method is used for expressions of the form \((a+b)^n\), and the Binomial Theorem provides a formula for expanding these expressions into a sum of terms. The general formula is: \[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\] Each term in the expansion involves a binomial coefficient, \( \binom{n}{k} \), which signifies the number of ways to choose \(k\) items from \(n\). In our example, we expanded \((x + \frac{1}{x})^{40}\). By using the Binomial Theorem, each term is generated based on its position \(k\) in the expansion, allowing fast computation of large powers in a polynomial form.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations such as addition and multiplication. In the context of our exercise, \((x + \frac{1}{x})\) is an expression composed of two terms. Raising it to the 40th power complicates it as we involve both powers of \(x\) and rational terms like \(\frac{1}{x}\).
Using a step-by-step approach, each contributed term can be identified within the larger expression. Understanding how to manipulate these elements according to algebraic rules is fundamental. Terms like \(x^{40}\) or \(40x^{38}\) arise through a series of operations dictated by the Binomial Theorem, where each variable is raised to specific powers based on the term’s position.

Combinatorics

Combinatorics is the branch of mathematics dealing with combinations of objects. It's crucial in calculating binomial coefficients, which determine the number of ways to choose elements from a set. In our expansion problem, each coefficient, \(\binom{40}{k}\), is calculated using combinations.

• For the first term, \(\binom{40}{0} = 1\), indicating a single way to choose 0 elements from 40.
• The second term's coefficient, \(\binom{40}{1}\), equals 40, showing 40 ways to choose 1 out of 40.
• Likewise, \(\binom{40}{2} = 780\) reflects the 780 combinations to select 2 elements from 40.

Understanding and calculating these coefficients are key in simplifying complex binomial expansions. Through combinatorial principles, these coefficients tell us how each term contributes to the overarching expression.