Problem 11 - Algebra and Trigonometry

Question

\(1-12\) . Use Pascal's triangle to expand the expression. $$ \left(\frac{1}{x}-\sqrt{x}\right)^{5} $$

Step-by-Step Solution

Answer

\(\left(\frac{1}{x} - \sqrt{x}\right)^5 = \frac{1}{x^5} - \frac{5}{x^{7/2}} + \frac{10}{x^2} - \frac{10}{x^{1/2}} + 5x - x^{5/2}\).

Step 1: Identify the Row in Pascal's Triangle

For the expansion of \((a+b)^n\), where \(n = 5\), we use the 6th row of Pascal's Triangle, which is \(1, 5, 10, 10, 5, 1\). Each of these coefficients will be used for the terms in the expansion.

Step 2: Write Down the General Term Formula

The general term in the expansion of \((a+b)^n\) is given by \(\binom{n}{k}a^{n-k}b^k\). Here, \(a = \frac{1}{x}\), \(b = -\sqrt{x}\), and \(n = 5\).

Step 3: Expand Using the Coefficients from Pascal's Triangle

Using the coefficients \(1, 5, 10, 10, 5, 1\), we write the expansion as:\[\left(\frac{1}{x} - \sqrt{x}\right)^5 = \binom{5}{0}\left(\frac{1}{x}\right)^5(-\sqrt{x})^0 + \binom{5}{1}\left(\frac{1}{x}\right)^4(-\sqrt{x})^1 + \binom{5}{2}\left(\frac{1}{x}\right)^3(-\sqrt{x})^2 + \binom{5}{3}\left(\frac{1}{x}\right)^2(-\sqrt{x})^3 + \binom{5}{4}\left(\frac{1}{x}\right)^1(-\sqrt{x})^4 + \binom{5}{5}\left(\frac{1}{x}\right)^0(-\sqrt{x})^5\]

Step 4: Simplify Each Term

We'll simplify each term one by one:1. \(\binom{5}{0}\left(\frac{1}{x}\right)^5 = \frac{1}{x^5}\).2. \(\binom{5}{1}\left(\frac{1}{x}\right)^4(-\sqrt{x}) = -\frac{5}{x^4}x^{1/2} = -\frac{5}{x^{7/2}}\).3. \(\binom{5}{2}\left(\frac{1}{x}\right)^3(-\sqrt{x})^2 = \frac{10}{x^3}x = \frac{10}{x^2}\).4. \(\binom{5}{3}\left(\frac{1}{x}\right)^2(-\sqrt{x})^3 = -\frac{10}{x^2}x^{3/2} = -\frac{10}{x^{1/2}}\).5. \(\binom{5}{4}\left(\frac{1}{x}\right)(-\sqrt{x})^4 = \frac{5}{x}x^2 = 5x\).6. \(\binom{5}{5}(-\sqrt{x})^5 = -x^{5/2}\).

Step 5: Combine the Simplified Terms

The final expansion of \(\left(\frac{1}{x} - \sqrt{x}\right)^5\) is:\[\frac{1}{x^5} - \frac{5}{x^{7/2}} + \frac{10}{x^2} - \frac{10}{x^{1/2}} + 5x - x^{5/2}\]

Key Concepts

Binomial Expansion

The binomial expansion is a powerful algebraic tool used to expand expressions raised to a power. When you have an expression like \((a+b)^n\), the binomial expansion provides a way to express it as a sum of terms. Each term includes specific coefficients and powers of the variables involved. This method is particularly useful for large powers and can simplify complex algebraic problems. The key feature of the binomial expansion is that it transforms a complex power expression into a series of manageable terms. For example, in our original exercise, we expand the expression \(\left(\frac{1}{x} - \sqrt{x}\right)^5\) using this method. The binomial theorem helps in systematically identifying each term by combining powers of the individual components with the respective binomial coefficients from Pascal's Triangle.

Binomial Coefficients

In a binomial expansion, binomial coefficients play a vital role as they determine the weights of each term in the expanded expression. These coefficients can be directly derived from Pascal's Triangle, which is a triangular array of numbers. Each row of Pascal’s Triangle corresponds to the coefficients of an expanded binomial expression raised to a specific power. For instance, for the power \(5\), we refer to the 6th row of Pascal’s Triangle: \(1, 5, 10, 10, 5, 1\). These numbers are precisely the coefficients for each term in the expansion of \((a + b)^5\), dictating how the components \(a\) and \(b\) combine. The coefficients ensure that the expansion is accurately weighted and mirror the symmetrical nature of Pascal’s Triangle.

Simplifying Expressions

Simplifying expressions is a pivotal step in the process of binomial expansion since it allows for clarity and better understanding. After applying the binomial expansion, each term will include expressions that need further simplification for ease of interpretation. For example, complex fractions or expressions involving roots can be simplified to more concise forms. In our given problem, terms like \(\frac{10}{x^3}x\) became simplified to \(\frac{10}{x^2}\). This step involves applying algebraic rules such as combining like terms, performing operations with exponents, and simplifying fractions. Simplification reduces the expression to its easiest form, making it easier to understand and manipulate further if necessary. It is an essential skill in all areas of algebra and calculus.