Problem 30 - 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

The expanded expression is: \(x^{\frac{5}{2}} + 5x^{\frac{3}{2}} + 10x^{\frac{1}{2}} + 10x^{-\frac{1}{2}} + 5x^{-\frac{3}{2}} + x^{-\frac{5}{2}}\).

Step 1: Identify the Binomial Components

The binomial theorem states: \[ (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]Here, identify the components as \( a = \sqrt{x} \), \( b = \frac{1}{\sqrt{x}} \), and \( n = 5 \).

Step 2: Write the Binomial Expansion

Use the binomial theorem formula to begin expanding:\[ (\sqrt{x} + \frac{1}{\sqrt{x}})^{5} = \sum_{k=0}^{5} \binom{5}{k} (\sqrt{x})^{5-k} \left(\frac{1}{\sqrt{x}}\right)^k \]

Step 3: Calculate Each Term

Now calculate each term using the formula:- For \( k=0 \): \[ \binom{5}{0} = 1, \quad (\sqrt{x})^{5} \cdot 1 = x^{\frac{5}{2}} \] - For \( k=1 \): \[ \binom{5}{1} = 5, \quad (\sqrt{x})^{4} \cdot \frac{1}{\sqrt{x}} = x^{\frac{3}{2}} \] - For \( k=2 \): \[ \binom{5}{2} = 10, \quad (\sqrt{x})^{3} \cdot \frac{1}{x} = x^{\frac{1}{2}} \] - For \( k=3 \): \[ \binom{5}{3} = 10, \quad (\sqrt{x})^{2} \cdot \frac{1}{x^{\frac{3}{2}}} = x^{-\frac{1}{2}} \] - For \( k=4 \): \[ \binom{5}{4} = 5, \quad \sqrt{x} \cdot \frac{1}{x^{2}} = x^{-rac{3}{2}} \] - For \( k=5 \): \[ \binom{5}{5} = 1, \quad \frac{1}{x^{\frac{5}{2}}} = x^{-\frac{5}{2}} \]

Step 4: Combine and Simplify the Expansion

Combine the terms calculated in the previous step:\[ x^{\frac{5}{2}} + 5x^{\frac{3}{2}} + 10x^{\frac{1}{2}} + 10x^{-\frac{1}{2}} + 5x^{-\frac{3}{2}} + x^{-\frac{5}{2}} \]

Step 5: Write the Final Simplified Expression

The final expanded and simplified expression is:\[ x^{\frac{5}{2}} + 5x^{\frac{3}{2}} + 10x^{\frac{1}{2}} + 10x^{-\frac{1}{2}} + 5x^{-\frac{3}{2}} + x^{-\frac{5}{2}} \]

Key Concepts

Binomial Expansion

The binomial expansion is a powerful algebraic tool that allows us to expand expressions raised to a power. Originally formulated as the Binomial Theorem, it is a concise way to handle expressions that can be broken down into two distinct terms raised to a power, \( (a+b)^n \). The theorem gives us a sum of terms of the form \( \binom{n}{k} a^{n-k} b^k \), where \( \binom{n}{k} \) is the binomial coefficient. This coefficient represents the number of ways to choose \( k \) items from \( n \), and it can be calculated using the formula: \( \frac{n!}{k!(n-k)!} \).
For our example, the expansion of \( (\sqrt{x} + \frac{1}{\sqrt{x}})^5 \) involves finding these coefficients and carefully combining the terms raised to different powers, starting from zero up to five. Each term in the expansion will have a unique binomial coefficient, forming part of a symmetric polynomial.
The significance of the binomial expansion is its ability to simplify complex polynomials, making them easier to handle in further calculations or applications.

Exponents

In this exercise, exponents play a crucial role in the binomial expansion process. Exponents represent repeated multiplication and are expressed as a small number placed above and immediately to the right of the base number, as in \( x^2 \), meaning \( x \) multiplied by itself.
When dealing with terms like \( (\sqrt{x})^{n-k} \) and \( \left(\frac{1}{\sqrt{x}}\right)^k \), it is important to understand how exponents work, including how they behave in multiplication and division. Multiplying powers with the same base leads us to add the exponents, while dividing them involves subtraction. So, when you see terms like \( (\sqrt{x})^3 \cdot \frac{1}{x} = x^{\frac{1}{2}} \), exponents are combined using these rules.
Knowing how to manipulate exponents is essential to correctly expanding the binomial and simplifying the expression afterward. This understanding forms a backbone of algebraic calculations, particularly in polynomial expressions.

Polynomial Simplification

Simplifying polynomials involves combining like terms, where we collect terms with the same base and exponent into a single term. After expanding using the Binomial Theorem, you often end up with multiple terms that could initially seem complex. These must be organized to achieve a simplified, more compact expression.
In our example, after computing each term separately, we get: \( x^{\frac{5}{2}} + 5x^{\frac{3}{2}} + 10x^{\frac{1}{2}} + 10x^{-\frac{1}{2}} + 5x^{-\frac{3}{2}} + x^{-\frac{5}{2}} \). Here, no further simplification by combining terms is possible since each term has a distinct exponent. The most vital step is ensuring all terms are independently calculated and correctly ordered.
Polynomial simplification reduces the risk of errors in subsequent calculations. It allows for a cleaner presentation and clearer insight into the behavior of the polynomial, ensuring that core algebraic operations are not unnecessarily complicated.