Problem 15 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

Expanded expression: \( \frac{1}{x^5} - \frac{5\sqrt{x}}{x^4} + \frac{10}{x^3} - \frac{10\sqrt{x}}{x^2} + \frac{5}{x} - x^{5/2} \).

Step 1: Understand the Expression

The expression to expand is \( \left( \frac{1}{x} - \sqrt{x} \right)^5 \). This means we need to find the coefficients and powers for each term in the expansion using the binomial theorem.

Step 2: Use the Binomial Theorem

The binomial theorem states that \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). Here, \(a = \frac{1}{x}\), \(b = -\sqrt{x}\), and \(n = 5\).

Step 3: Identify Coefficients Using Pascal's Triangle

The coefficients for the expansion are given by the 6th row of Pascal’s triangle: 1, 5, 10, 10, 5, 1.

Step 4: Calculate Each Term

For each term, use the formula: \( \binom{5}{k} \left( \frac{1}{x} \right)^{5-k} (-\sqrt{x})^k \).- For \(k = 0\): \( \binom{5}{0} \left( \frac{1}{x} \right)^5 = \frac{1}{x^5} \)- For \(k = 1\): \( \binom{5}{1} \left( \frac{1}{x} \right)^4 (-\sqrt{x}) = -\frac{5}{x^4}\sqrt{x} \)- For \(k = 2\): \( \binom{5}{2} \left( \frac{1}{x} \right)^3 (\sqrt{x})^2 = \frac{10}{x^3}\)- For \(k = 3\): \( \binom{5}{3} \left( \frac{1}{x} \right)^2 (-\sqrt{x})^3 = -\frac{10}{x^2}\sqrt{x}\)- For \(k = 4\): \( \binom{5}{4} \left( \frac{1}{x} \right)^1 (\sqrt{x})^4 = \frac{5}{x}\)- For \(k = 5\): \( \binom{5}{5} (\sqrt{x})^5 = -x^{5/2} \)

Step 5: Combine All Terms

Combine all the individual terms derived from Step 4 to write the expanded expression:\[ \frac{1}{x^5} - \frac{5\sqrt{x}}{x^4} + \frac{10}{x^3} - \frac{10\sqrt{x}}{x^2} + \frac{5}{x} - x^{5/2}. \]

Key Concepts

Binomial Theorem

The Binomial Theorem is a fundamental principle in algebra that provides a way to expand expressions that are raised to a power. It can be written as \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). In this formula:

• \(a\) and \(b\) are any real or complex numbers or expressions.
• \(n\) is a non-negative integer indicating the power to which the binomial is raised.
• The sigma notation \(\sum\) indicates that we sum all terms from \(k = 0\) to \(k = n\).
• \(\binom{n}{k}\) represents the binomial coefficient, which can be calculated as \(\frac{n!}{k!(n-k)!}\).

The theorem helps to break down a complex expression into simpler monomial terms, each with a specific binomial coefficient. For our expression \(\left(\frac{1}{x} - \sqrt{x}\right)^5\), the Binomial Theorem provides the framework to express this as a sum of terms with defined powers and coefficients.

Expression Expansion

Expression expansion involves expressing a power of a binomial as a sum of individual terms. Using the Binomial Theorem, each term in the expansion of \(\left(\frac{1}{x} - \sqrt{x}\right)^5\) can be calculated systematically:

• Determine the terms from the expansion formula \(\binom{n}{k} a^{n-k} b^k\).
• For each term, identify the power of each component: \((\frac{1}{x})^{5-k}\) and \((-\sqrt{x})^k\).
• Multiply by the coefficient from Pascal's Triangle.

Each term is unique, derived from the interaction of \(\frac{1}{x}\) and \(-\sqrt{x}\), adjusted by their location in the sequence (sign, numerator/denominator appearance). Finally, you combine all the terms to complete the expansion, forming an algebraic expression that portrays the expanded form of the entire binomial.

Pascal's Triangle Coefficients

Pascal's Triangle is a triangular array of numbers, where each number is the sum of the two directly above it. It's an efficient way to find the binomial coefficients \(\binom{n}{k}\) needed for binomial expansion. Each row corresponds to an exponent \(n\) in the binomial expansion.
To expand \(\left(\frac{1}{x} - \sqrt{x}\right)^5\), we refer to the 6th row of Pascal's Triangle, which contains the coefficients: 1, 5, 10, 10, 5, 1. These coefficients:

• Dictate the magnitude of each term in the expanded expression.
• Help simplify expression handling, avoiding tedious calculations for coefficients each time.

Using Pascal's Triangle not only streamlines binomial expansion but also visually represents how these coefficients build upon one another, illustrating the symmetry and pattern present in algebraic expansions.