Problem 29 - Algebra and Trigonometry with Analytic Geometry

Question

Use the binomial theorem to expand and simplify. $$ \left(\sqrt{x}-\frac{1}{\sqrt{x}}\right)^{5} $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Binomial Theorem

The binomial theorem provides a way to expand expressions of the form \((a + b)^n\). It states that \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\), where \(\binom{n}{k}\) is the binomial coefficient.

Step 2: Identify Variables

Here, we have \((a, b, n) = (\sqrt{x}, -\frac{1}{\sqrt{x}}, 5)\). We will use these values in the binomial expansion.

Step 3: Compute Binomial Coefficients

The binomial coefficients for the expansion are given as \(\binom{5}{0}, \binom{5}{1}, \binom{5}{2}, \binom{5}{3}, \binom{5}{4}, \binom{5}{5}\). These are 1, 5, 10, 10, 5, and 1 respectively.

Step 4: Apply Binomial Theorem to Expand

Substitute each term in the expression: \((\sqrt{x} - \frac{1}{\sqrt{x}})^5 = \sum_{k=0}^{5} \binom{5}{k} (\sqrt{x})^{5-k} \left(-\frac{1}{\sqrt{x}}\right)^k\).

Step 5: Calculate Each Term of the Expansion

Calculate each term using the expressions obtained: - \((\sqrt{x})^5 = x^{5/2}\)- \((\sqrt{x})^4(-\frac{1}{\sqrt{x}}) = -x^{3/2}\times\frac{1}{\sqrt{x}} = -x\)- \((\sqrt{x})^3(-\frac{1}{\sqrt{x}})^2 = x^{3/2}\times\frac{1}{x} = x^{1/2}\)- \((\sqrt{x})^2(-\frac{1}{\sqrt{x}})^3 = -x\times (-1) = -x^{-1/2}\)- \((\sqrt{x})(-\frac{1}{\sqrt{x}})^4 = (-1)\times (-1) = x^{-1}\)- \((-\frac{1}{\sqrt{x}})^5 = -\frac{1}{x^{5/2}}\).

Step 6: Multiply by Binomial Coefficients and Combine

Combine all the terms:\[\begin{align*}&= 1\times x^{5/2} + 5\times(-x) + 10\times x^{1/2} + 10\times(-x^{-1/2}) + 5\times(-x^{-1}) + 1\times(-x^{-5/2}) \&= x^{5/2} - 5x + 10x^{1/2} - 10x^{-1/2} - 5x^{-1} - x^{-5/2}\end{align*}\]

Step 7: Result of the expansion

The expansion of \((\sqrt{x} - \frac{1}{\sqrt{x}})^5\) simplifies to: \[ x^{5/2} - 5x + 10x^{1/2} - 10x^{-1/2} - 5x^{-1} - x^{-5/2} \]

Key Concepts

Binomial Coefficients

Binomial coefficients are fundamental to understanding the binomial theorem. They represent the number of ways to choose a subset of items from a larger set, specifically "n choose k", often denoted as \( \binom{n}{k} \). These coefficients appear when expanding an expression of the form \((a + b)^n\) and are calculated using the formula:\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]where \( n! \) (n factorial) is the product of all positive integers up to \( n \). In our exercise, we needed the coefficients for the expansion where \( n = 5 \). Thus, the coefficients were \( 1, 5, 10, 10, 5, \) and \( 1 \). These numbers correspond to each term in the expansion, providing a multiplier that, combined with powers of the variables, forms each term in the series. Understanding these coefficients is crucial for evaluating how each component contributes to the entire polynomial's structure.

Expansion of Binomials

The expansion of binomials refers to the process of expressing a power of a binomial as a sum of terms using the binomial theorem. The theorem itself allows us to expand binomials of the form \((a + b)^n\) into a series. For our problem, we expanded \((\sqrt{x} - \frac{1}{\sqrt{x}})^5\) using this theorem:\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]By identifying \(a\) as \(\sqrt{x}\) and \(b\) as \(-\frac{1}{\sqrt{x}}\), and \(n\) as 5, we could systematically replace these variables and compute individual terms. Each term in the series is derived by altering the powers of \(a\) and \(b\), while multiplying by the respective binomial coefficient. This method breaks down the polynomial into a manageable format, revealing the specific contributions of each element.

Simplification of Polynomials

Simplification of polynomials involves combining and reducing expressions to a form that is easier to interpret or more practical for calculations. Once we have expanded a polynomial using the binomial theorem, the next step is to simplify the resulting expression.In this exercise, after using the binomial theorem, the expression became:\[ x^{5/2} - 5x + 10x^{1/2} - 10x^{-1/2} - 5x^{-1} - x^{-5/2} \]The key to simplification involves acknowledging like terms and ensuring every combination of powers is calculated accurately. Notice, each term here already is distinct by power, indicating no further simplification through combining like terms is possible. The simplification process is crucial for making complex expressions more tractable for further mathematical operations or applications.