Problem 26 - Algebra and Trigonometry

Question

Expand each expression using the Binomial Theorem. $$ (\sqrt{x}-\sqrt{3})^{4} $$

Step-by-Step Solution

Answer

The expansion is \( x^2 - 4 x^{3/2} \sqrt{3} + 18 x - 12 x^{1/2} \sqrt{3} + 81 \).

Step 1: State the Binomial Theorem

The Binomial Theorem states that \[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \] where \( \binom{n}{k} \) is a binomial coefficient.

Step 2: Identify the terms

In this case, let \( a = \sqrt{x} \), \( b = -\sqrt{3} \), and \( n = 4 \). We will substitute these values into the binomial theorem.

Step 3: Calculate binomial coefficients

For each term in the expansion, use the binomial coefficient \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \), where \( n = 4 \). The coefficients are: \( \binom{4}{0} = 1 \), \( \binom{4}{1} = 4 \), \( \binom{4}{2} = 6 \), \( \binom{4}{3} = 4 \), and \( \binom{4}{4} = 1 \).

Step 4: Write out each term in the expansion

Substitute the values into the equation: \[ (\sqrt{x} - \sqrt{3})^4 = \binom{4}{0} (\sqrt{x})^{4} (-\sqrt{3})^0 + \binom{4}{1} (\sqrt{x})^{3} (-\sqrt{3})^1 + \binom{4}{2} (\sqrt{x})^{2} (-\sqrt{3})^2 + \binom{4}{3} (\sqrt{x})^{1} (-\sqrt{3})^3 + \binom{4}{4} (\sqrt{x})^{0} (-\sqrt{3})^4 \]

Step 5: Simplify each term

Simplify the powers and products: 1. \( \binom{4}{0} (\sqrt{x})^{4} (-\sqrt{3})^0 = x^2 \)2. \( \binom{4}{1} (\sqrt{x})^{3} (-\sqrt{3})^1 = -4 x^{3/2} \sqrt{3} \)3. \( \binom{4}{2} (\sqrt{x})^{2} (-\sqrt{3})^2 = 6 x (3) = 18 x \)4. \( \binom{4}{3} (\sqrt{x})^{1} (-\sqrt{3})^3 = -4 x^{1/2} (3 \sqrt{3}) = -12 x^{1/2} \sqrt{3} \)5. \( \binom{4}{4} (\sqrt{x})^{0} (-\sqrt{3})^4 = 81 \)

Step 6: Combine all simplified terms

Add all the simplified terms together to get the final expanded form: \[ (\sqrt{x} - \sqrt{3})^4 = x^2 - 4 x^{3/2} \sqrt{3} + 18 x - 12 x^{1/2} \sqrt{3} + 81 \]

Key Concepts

Binomial Expansion

The binomial expansion is a powerful method to expand expressions of the form \( (a + b)^n \). This theorem allows us to express these terms as a sum of individual terms, involving binomial coefficients, powers of \(a \) and \(b \). By using the formula: \[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \], we systematically break down the problem. Here, \binom{n}{k} \ represents the binomial coefficient.

Binomial Coefficient

The binomial coefficient \( \binom{n}{k} \) is a crucial component of the binomial theorem. This coefficient is 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've seen binomial coefficients like \( \binom{4}{0}=1 \), \( \binom{4}{1}=4 \), \( \binom{4}{2}=6 \), \( \binom{4}{3}=4 \), and \( \binom{4}{4}=1 \). Each coefficient guides the expansion of each respective term in the polynomial.

Simplifying Expressions

Simplifying the resultant expressions is the last, but an important, step in using the binomial theorem. Once substituted into the binomial expansion formula, each term involves operations such as exponents and multiplication. For example, in the term \( \binom{4}{3}(\sqrt{x})^{1}(-\sqrt{3})^3 \), simplifying gives \( -12 x^{1/2} \sqrt{3} \). Always simplify systematically: handle exponents first, then process any multiplications. Combining these terms after simplifying all individual terms yields the final expanded expression.

Algebraic Expressions

An algebraic expression formed through binomial expansion typically includes variables, coefficients, and exponents. From our exercise, the initial algebraic expression \( (\sqrt{x} - \sqrt{3})^4 \) turns into multiple terms involving various powers of \( x \) and constants. Recognizing how to manage and interpret these expressions is foundational in algebra. By mastering the binomial theorem, one becomes adept at deconstructing complex polynomial expressions into manageable components.