Problem 22 - College Algebra

Question

For the following exercises, use the Binomial Theorem to expand each binomial. $$ (\sqrt{x}-\sqrt{y})^{5} $$

Step-by-Step Solution

Answer

\[x^{5/2} - 5 x^2 y^{1/2} + 10 x^{3/2} y - 10 x y^{3/2} + 5 x^{1/2} y^2 - y^{5/2}\]

Step 1: Identify the Formula

The binomial theorem states that \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). Identify \(a\), \(b\), and \(n\) for the given binomial expression \((\sqrt{x} - \sqrt{y})^5\) as \(a = \sqrt{x}\), \(b = -\sqrt{y}\), and \(n = 5\).

Step 2: Calculate the Binomial Coefficients

For each term in the expansion, we need the binomial coefficient \(\binom{n}{k}\). We calculate these coefficients for \(n = 5\) and \(k = 0, 1, 2, 3, 4, 5\). These are \(\binom{5}{0} = 1\), \(\binom{5}{1} = 5\), \(\binom{5}{2} = 10\), \(\binom{5}{3} = 10\), \(\binom{5}{4} = 5\), and \(\binom{5}{5} = 1\).

Step 3: Expand Using the Binomial Theorem

Write out each term using the binomial coefficients and the formula: \(\binom{5}{k} (\sqrt{x})^{5-k} (-\sqrt{y})^k\) for \(k = 0, 1, 2, 3, 4, 5\). This gives the terms: \(1 \cdot (\sqrt{x})^5 \cdot (-\sqrt{y})^0\), \(5 \cdot (\sqrt{x})^4 \cdot (-\sqrt{y})^1\), \(10 \cdot (\sqrt{x})^3 \cdot (-\sqrt{y})^2\), \(10 \cdot (\sqrt{x})^2 \cdot (-\sqrt{y})^3\), \(5 \cdot (\sqrt{x})^1 \cdot (-\sqrt{y})^4\), \(1 \cdot (\sqrt{x})^0 \cdot (-\sqrt{y})^5\).

Step 4: Compute Each Term

Simplify each term:1. \((\sqrt{x})^5 = x^{5/2}\)2. \(-5(\sqrt{x})^4(\sqrt{y}) = -5x^2y^{1/2}\)3. \(10(\sqrt{x})^3(\sqrt{y})^2 = 10x^{3/2}y\)4. \(-10(\sqrt{x})^2(\sqrt{y})^3 = -10x y^{3/2}\)5. \(5\sqrt{x}(\sqrt{y})^4 = 5x^{1/2}y^2\)6. \(-(\sqrt{y})^5 = -y^{5/2}\).

Step 5: Combine All Terms

Combine all the computed terms to get the expanded form: \[x^{5/2} - 5 x^2 y^{1/2} + 10 x^{3/2} y - 10 x y^{3/2} + 5 x^{1/2} y^2 - y^{5/2}.\]

Key Concepts

Binomial Expansion

The concept of binomial expansion is central to understanding expressions raised to powers, especially when dealing with expressions like \[(a + b)^n\]where binomial coefficients and powers of terms come into play. To expand a binomial expression like \[(\sqrt{x} - \sqrt{y})^5,\]we use the Binomial Theorem, which provides a systematic way of expanding expressions based on the pattern of coefficients and powers. The principle is grounded in extending the simple operations of algebra to a more complex power of a binomial.To apply this:

• Recognize the terms of the binomial, which here are \(\sqrt{x}\) and \(-\sqrt{y}.\)
• The power to which the binomial is raised, known as \(n,\) is 5 in this case.

The theorem outlines that each term in the expansion comes from the added products of the binomial terms raised to different powers, multiplied by specific coefficients called "binomial coefficients." This method simplifies the task of raising binomials to larger powers, circumventing tedious multiplication. Depending on the value of \(n,\) the expansion yields \(n+1\) terms.

Binomial Coefficients

Key to correctly applying the binomial theorem are the so-called binomial coefficients, represented by \[\binom{n}{k},\]which articulate how terms in the expansion are weighted. They are calculated using the formula:\[\binom{n}{k} = \frac{n!}{k!(n-k)!},\]where \(!\) represents factorial, and \(n\) and \(k\) are integers with \(0 \leq k \leq n.\)In the context of our problem expanding \((\sqrt{x} - \sqrt{y})^5,\)we calculate these coefficients for \(n = 5\) and specific \(k\) values ranging from 0 to 5:

• \(\binom{5}{0} = 1\)
• \(\binom{5}{1} = 5\)
• \(\binom{5}{2} = 10\)
• \(\binom{5}{3} = 10\)
• \(\binom{5}{4} = 5\)
• \(\binom{5}{5} = 1\)

Each coefficient plays a crucial role in shaping the individual terms in the expansion by scaling the powers of the binomial components.

Algebraic Expressions

Dealing with algebraic expressions involves manipulating symbols and numbers to express quantities, solve equations, or represent ideas. An expression like \((\sqrt{x} - \sqrt{y})^5\)demands recognizing the components and how they interact under operations like multiplication and exponentiation. As part of binomial expansion, each term within \((a + b)^n\)is an algebraic expression comprising factors of \(a\) and \(b\) raised to particular powers determined by their position in the series.For example, terms like \[x^{5/2}\] and \[-5x^2y^{1/2}\]are results of the calculated aspects of the expression, where the focus is on:

• Correctly applying radicals and powers from \(a\) and \(b.\)
• Combining like terms efficiently to streamline the expression.
• Ensuring negative bases are treated properly when they interact with exponents.

Through understanding these interactions, we can carefully simplify and manipulate algebraic expressions to achieve a desired form or solution.