Problem 3 - Algebra and Trigonometry

Question

\(1-12\) . Use Pascal's triangle to expand the expression. $$ \left(x+\frac{1}{x}\right)^{4} $$

Step-by-Step Solution

Answer

The expansion is \( x^4 + 4x^2 + 6 + \frac{4}{x^2} + \frac{1}{x^4} \).

Step 1: Identify the row in Pascal's Triangle

To expand \( \left( x + \frac{1}{x} \right)^4 \), we need to identify the coefficients from Pascal's Triangle for the row corresponding to the power of 4. The coefficients are: 1, 4, 6, 4, 1.

Step 2: Write the general expansion formula

The binomial expansion formula is given by: \( (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \). Here, \( a = x \), \( b = \frac{1}{x} \), and \( n = 4 \).

Step 3: Expand using the coefficients

Using the coefficients from the 4th row of Pascal's Triangle (1, 4, 6, 4, 1), we expand \( \left( x + \frac{1}{x} \right)^4 \): \[ (x + \frac{1}{x})^4 = 1 \cdot x^4 + 4 \cdot x^3 \cdot \frac{1}{x} + 6 \cdot x^2 \cdot \left(\frac{1}{x}\right)^2 + 4 \cdot x \cdot \left(\frac{1}{x}\right)^3 + 1 \cdot \left(\frac{1}{x}\right)^4 \]

Step 4: Simplify each term in the expansion

Calculate each term individually: - \( 1 \cdot x^4 = x^4 \)- \( 4 \cdot x^3 \cdot \frac{1}{x} = 4x^2 \)- \( 6 \cdot x^2 \cdot \left(\frac{1}{x}\right)^2 = 6 \)- \( 4 \cdot x \cdot \left(\frac{1}{x}\right)^3 = 4 \cdot \frac{1}{x^2} = \frac{4}{x^2} \)- \( 1 \cdot \left(\frac{1}{x}\right)^4 = \frac{1}{x^4} \)

Step 5: Write the final expanded form

Combine all simplified terms to write the final expanded form of \( \left(x + \frac{1}{x}\right)^4 \): \[ x^4 + 4x^2 + 6 + \frac{4}{x^2} + \frac{1}{x^4} \]

Key Concepts

Understanding Binomial Expansion

Binomial expansion is a technique used to expand expressions of the form \((a + b)^n\), where \(n\) is a non-negative integer. This involves expressing the power of a binomial as a sum of multiple terms, each term being a product of specific coefficients, powers of \(a\), and powers of \(b\). The coefficients in this expansion come from Pascal's Triangle, which provides a straightforward method to determine them.

When we deal with problems like \((x + \frac{1}{x})^4\), binomial expansion allows us to break it down into a sum of terms, making it easier to solve or simplify the expression. Using the formula for binomial expansion \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\), you can systematically calculate each term. These steps are crucial for handling complicated expressions in algebra.

Exploring Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols. It is a unifying thread of almost all of mathematics and is used to describe relationships and changes that occur over time.

In our exercise, algebra helps us work with expressions containing variables like \(x\) and \(\frac{1}{x}\). By applying algebraic rules, we can manipulate and expand these expressions. This expansion process makes use of operations like multiplication, division, exponentiation, and simplification to break down complex relationships into understandable parts. Algebra forms the foundation for more advanced topics, including polynomial expansion, helping us solve numerous real-world problems.

Breaking Down Polynomial Expansion

Polynomial expansion involves writing a polynomial, which is an expression made up of variables and coefficients, in an expanded form. When we perform polynomial expansion, especially using the binomial theorem, each expanded term appears as a power product of the original variables.

In the context of the given exercise, \((x + \frac{1}{x})^4\), the polynomial is expanded into several terms: \(x^4\), \(4x^2\), \(6\), \(\frac{4}{x^2}\), and \(\frac{1}{x^4}\). This transformed expression is a polynomial, where each term involves a degree of \(x\) or its reciprocal. By expanding polynomials, we're able to see the individual components and how they relate to the original expression, thus making complex algebraic structures easier to handle and understand.