Problem 32 - Precalculus: Mathematics for Calculus

Question

Find the indicated terms in the expansion of the given binomial. 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: Analyze the Binomial Expression

The expression given is \(\left(x + \frac{1}{x}\right)^{40}\). This is a binomial expression where each term in the expansion can be represented using the Binomial Theorem.

Step 2: Apply the Binomial Theorem

The Binomial Theorem states that \((a+b)^n\) can be expanded as a series \(\sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k}\). Here, \(a = x\), \(b = \frac{1}{x}\), and \(n = 40\).

Step 3: Find the First Term

The first term \(T_0\) in the expansion is obtained when \(k = 0\): \(\binom{40}{0} x^{40-0} \left(\frac{1}{x}\right)^{0} = x^{40} \).

Step 4: Find the Second Term

The second term \(T_1\) in the expansion is when \(k = 1\): \(\binom{40}{1} x^{40-1} \left(\frac{1}{x}\right)^{1} = 40x^{39}\left(\frac{1}{x}\right) = 40x^{38} \).

Step 5: Find the Third Term

The third term \(T_2\) in the expansion is when \(k = 2\): \(\binom{40}{2} x^{40-2} \left(\frac{1}{x}\right)^{2} = \frac{40 \times 39}{2} x^{38} \left(\frac{1}{x^2}\right) = 780 x^{36} \).

Key Concepts

Binomial Expansion

The binomial expansion is a method to expand expressions that are raised to a particular power. In the expression \[ (a + b)^n \],where \(a\) and \(b\) are any numbers or algebraic symbols, and \(n\) is a positive integer, the expansion includes several terms that can be calculated using the Binomial Theorem. This theorem provides a systematic way to expand expressions of this form.

Applying the Binomial Theorem, we find the entire expansion of an expression like \(\left( x + \frac{1}{x} \right)^{40}\).The theorem tells us how each term in this expansion can be derived from the formula:- The general term is given by \(\binom{n}{k} a^{n-k} b^{k}\).- The expression expands to a series of terms like \(T_0, T_1, T_2, \ldots\).
By substituting values into the formula, each term is calculated one by one, considering that \(k\) ranges from 0 to \(n\). The terms are obtained by substituting different values of \(k\), which denotes the specific term's number in the sequence.

Binomial Coefficients

Binomial coefficients play a crucial role in binomial expansions. They are represented as \(\binom{n}{k}\),which is read as \("n\) choose \(k\)." These coefficients determine the weight of each term in the expansion.

The formula for calculating binomial coefficients is:\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]where \(n!\) (n factorial) is the product of all positive integers from 1 to \(n\). This calculation gives the number of ways to choose \(k\) items from \(n\) without considering order.- For example, in the expression \(\left(x+\frac{1}{x}\right)^{40}\),the binomial coefficients for the first few terms are: - \(\binom{40}{0} = 1\) - \(\binom{40}{1} = 40\) - \(\binom{40}{2} = 780\)
Binomial coefficients are central to understanding how the terms in the expansion grow and shift as you move from term to term. They help in distributing the resultant product across the expanded expression.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and arithmetic operations. In the binomial expansion \((x + \frac{1}{x})^{40}\),we deal with algebraic terms which include variables and their exponents.

Each term in a binomial expansion like \((x + \frac{1}{x})^{40}\)begins with an algebraic expression that is manipulated through processes like multiplication and exponentiation.- The power of the variables, such as \(x\) or \(\frac{1}{x}\), changes depending on their position in the expansion. This is determined by the binomial theorem through the formula \(a^{n-k} b^{k}\).- Here, the algebraic expression gets modified as exponents change from term to term.
During the expansion process, these expressions reveal the mathematical relationships and patterns. For example, derivations like \(x^{40}, 40x^{38}, 780x^{36},\)illustrate the progression of terms by diminishing the exponent of \(x\) while increasing the coefficient derived from binomial factors. Understanding these principles is key to solving problems efficiently using the binomial theorem.