Problem 7 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify Coefficients from Pascal's Triangle

Pascal's Triangle provides the coefficients for binomial expansion. For \((x + \frac{1}{x})^4\), look at the 5th row of Pascal's Triangle (since rows start at zero). The coefficients are 1, 4, 6, 4, 1.

Step 2: Write the General Expansion Formula

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

Step 3: Apply the Coefficients to Each Term

Use the coefficients from Step 1 to expand using the binomial formula: 1. First term: \(1 \cdot x^4 \cdot (\frac{1}{x})^0 = x^4\).2. Second term: \(4 \cdot x^3 \cdot (\frac{1}{x})^1 = 4x^2\).3. Third term: \(6 \cdot x^2\cdot (\frac{1}{x})^2 = 6\).4. Fourth term: \(4 \cdot x^1\cdot (\frac{1}{x})^3 = 4\frac{1}{x^2}\).5. Fifth term: \(1 \cdot x^0\cdot (\frac{1}{x})^4 = \frac{1}{x^4}\).

Step 4: Combine All the Terms

Combine all the terms obtained from Step 3 to form the expanded expression:\(x^4 + 4x^2 + 6 + \frac{4}{x^2} + \frac{1}{x^4}\).

Key Concepts

Pascal's Triangle

Pascal's Triangle is a simple yet powerful tool used to find coefficients in the binomial expansion of algebraic expressions. It is a triangular array of numbers where each number is the sum of the two directly above it. This structure makes it particularly useful for expanding binomials like

• (a + b)^2
• (a + b)^3
• (a + b)^4

For example, in our exercise, we used the 5th row of Pascal's Triangle, which contains the values 1, 4, 6, 4, and 1. This was crucial because, for the expansion of \((x + \frac{1}{x})^4\), these numbers serve as the coefficients for each term in the expansion sequence. The ease of using Pascal's Triangle lies in its intuitive layout, and once you understand its pattern, you can effortlessly determine coefficients for any binomial expansion. Simply choose the row corresponding to the power (n) of the expression, and you'll have your coefficients ready to use.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations (such as addition and multiplication) that are put together to express a specific mathematical idea or problem. The expression \((x + \frac{1}{x})^4\) is a notable example of an algebraic expression.
Since it's in the form of a binomial expression, it can be expanded using the binomial formula or methods such as Pascal's Triangle.
Each component of the algebraic expression holds significance:

• 'x' represents a variable that can take various values.
• '\(\frac{1}{x}\)' is the reciprocal of x, often featuring due to the inverse property in expressions.

Algebra focuses on understanding the relationships defined by expressions and equations. Knowing how to manipulate these expressions, like expanding binomials, is essential for solving a wide range of mathematical problems.

Polynomials

Polynomials are a specific type of algebraic expression composed of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents. A simple form of polynomial is the one-variable polynomial: \(ax^n + bx^{n-1} + \ldots + c\).
In the binomial expansion of \((x + \frac{1}{x})^4\), the result is a polynomial expression.
The expanded expression \(x^4 + 4x^2 + 6 + \frac{4}{x^2} + \frac{1}{x^4}\) shows a polynomial where the terms are powers of x, both positive and negative.

• The degree of this polynomial in terms of x is 4, which is the highest power of x present.
• The coefficients, obtained from the Pascal's Triangle, define the weight of each term.
• Polynomials are a core concept in algebra and are used in various fields to model complex systems and solve equations.

Understanding the role of polynomials and how they form gives us insights into solving mathematical expressions elegantly and effectively.