Binomial Expansion Of 1/√(4-x) And Its Approximations

Delving into the Binomial Expansion of 1/√(4-x)

Binomial expansion is a powerful tool in mathematics, especially in calculus and algebra, that allows us to express powers of binomials as sums of terms involving binomial coefficients. In this article, we'll be discussing how to find the first three terms, in ascending powers of x, of the binomial expansion of 1/√(4-x), providing each coefficient in its simplest form. This exploration will not only enhance your understanding of binomial expansion but also demonstrate its practical applications in approximating complex expressions. Understanding and manipulating binomial expansions is fundamental in various areas of mathematics, including calculus, statistics, and numerical analysis. The ability to expand expressions like 1/√(4-x) into a series allows us to approximate their values for small values of x, which is particularly useful when dealing with functions that are difficult to compute directly. Mastering these techniques provides a solid foundation for tackling more advanced mathematical problems and real-world applications, where approximations are often necessary due to the complexity of the functions involved. This detailed exploration will equip you with the knowledge and skills to confidently apply binomial expansion in diverse mathematical scenarios.

Unpacking the Problem

Before we dive into the solution, let's clearly understand the problem at hand. We're tasked with finding the first three terms of the binomial expansion of the expression 1/√(4-x). This means we need to express this function as a series in the form a + bx + cx², where a, b, and c are constants that we need to determine. The binomial expansion formula is our primary tool here, but we'll also need to apply some algebraic manipulation to get the expression into a suitable form. This involves recognizing that 1/√(4-x) can be rewritten using fractional exponents and then applying the binomial theorem. The binomial theorem provides a systematic way to expand expressions of the form (1 + x)^n, where n is any real number. The challenge lies in correctly identifying the components of our expression and applying the formula accurately. Once we have the expansion, we'll focus on extracting the first three terms, which correspond to the constant term, the linear term (coefficient of x), and the quadratic term (coefficient of x²). This process will require careful attention to detail and algebraic precision to ensure that the coefficients are calculated correctly and presented in their simplest form. Through this exercise, we aim to not only find the solution but also to deepen your understanding of how binomial expansion works and how it can be applied to a variety of mathematical problems.

Step-by-Step Solution

Let's break down the solution step by step.

1. Rewrite the Expression

The first crucial step in dealing with this problem is to rewrite the given expression, 1/√(4-x), in a more manageable form that aligns with the binomial expansion formula. The binomial theorem is typically applied to expressions in the form (1 + x)^n, where n can be any real number, including fractions and negative numbers. Our expression needs to be transformed into this format to effectively utilize the theorem. To begin, we can express the square root in the denominator as a power of -1/2. This gives us (4-x)^(-1/2). Next, we need to factor out the constant term, 4, from inside the parentheses. This is a critical step because the binomial theorem requires the first term inside the parentheses to be 1. Factoring out 4 gives us 4^(-1/2) * (1 - x/4)^(-1/2). Notice how we have now separated the expression into two parts: a constant term, 4^(-1/2), and a binomial expression in the desired form, (1 - x/4)^(-1/2). This manipulation allows us to apply the binomial theorem to the second part, while the constant term will simply be multiplied with the result. Simplifying 4^(-1/2) gives us 1/√(4), which equals 1/2. This means our expression now looks like (1/2) * (1 - x/4)^(-1/2). We are now perfectly set up to apply the binomial expansion formula, and the subsequent steps will involve carefully substituting the appropriate values into the formula and simplifying the resulting terms.

2. Apply the Binomial Theorem

Now that we've successfully rewritten the expression in the form (1/2) * (1 - x/4)^(-1/2), we can proceed to apply the binomial theorem. The binomial theorem provides a formula for expanding expressions of the form (1 + y)^n, where n is a real number. In our case, y is -x/4 and n is -1/2. The binomial theorem states that (1 + y)^n = 1 + ny + (n(n-1)/2!)y² + ... , where the ellipsis indicates that the series continues with higher powers of y. To find the first three terms of the expansion, we need to calculate the constant term, the term involving y (which corresponds to x in our original expression), and the term involving y² (which corresponds to x²). Let's substitute y = -x/4 and n = -1/2 into the formula. The first term is simply 1. The second term is ny = (-1/2)(-x/4) = x/8. The third term involves calculating (n(n-1)/2!)y², which is [(-1/2)(-1/2 - 1)/2](-x/4)² = [(-1/2)(-3/2)/2](x²/16) = (3/8)(x²/16) = 3x²/128. Therefore, the first three terms of the expansion of (1 - x/4)^(-1/2) are 1 + x/8 + 3x²/128. These terms represent the foundation of our binomial expansion, and in the next step, we will multiply these terms by the constant factor (1/2) to complete the expansion of the original expression.

3. Multiply by the Constant

Having obtained the first three terms of the expansion of (1 - x/4)^(-1/2) as 1 + x/8 + 3x²/128, the next step is to multiply these terms by the constant factor we factored out earlier, which was 1/2. This multiplication is essential to complete the binomial expansion of the original expression, 1/√(4-x). By distributing the 1/2 across each term, we ensure that the expansion accurately represents the original function. Multiplying 1/2 by the first term, 1, gives us 1/2. Multiplying 1/2 by the second term, x/8, results in x/16. Finally, multiplying 1/2 by the third term, 3x²/128, gives us 3x²/256. Therefore, the first three terms of the binomial expansion of 1/√(4-x) are 1/2 + x/16 + 3x²/256. These terms represent a polynomial approximation of the original function, valid for values of x close to zero. This process of multiplying by the constant factor is a crucial step in ensuring that the expansion is complete and accurate, and it highlights the importance of keeping track of all the components of the expression throughout the calculation. With this step completed, we have successfully found the first three terms of the binomial expansion, and we can now use this expansion to approximate the value of the function for small values of x.

4. State the First Three Terms

After performing all the necessary calculations, we can now clearly state the first three terms, in ascending powers of x, of the binomial expansion of 1/√(4-x). These terms are:

• 1/2 (the constant term)
• x/16 (the linear term)
• 3x²/256 (the quadratic term)

Thus, the binomial expansion begins as 1/√(4-x) ≈ 1/2 + x/16 + 3x²/256. These terms provide an approximation of the function for values of x close to zero. The more terms we include in the expansion, the more accurate the approximation becomes. However, for small values of x, the first few terms often provide a sufficiently accurate approximation. This result is crucial because it allows us to replace a complex function, such as 1/√(4-x), with a simpler polynomial expression, which can be easier to work with in various mathematical contexts. For example, this approximation can be used to estimate the value of the function at a specific point, to analyze its behavior near zero, or to perform calculations that would be difficult or impossible with the original function. The ability to find and use binomial expansions is a fundamental skill in mathematics, and this example demonstrates the power and versatility of this technique in approximating functions and solving problems.

Applications and Approximations

The binomial expansion we've derived, 1/√(4-x) ≈ 1/2 + x/16 + 3x²/256, is not just a mathematical curiosity; it has significant practical applications, particularly in approximating values of the function for small x. One of the primary uses of binomial expansions is in scenarios where directly calculating the value of a function is difficult or computationally expensive. By using the expansion, we can replace the original function with a polynomial, which is much easier to evaluate. For instance, consider trying to find the value of 1/√(4-0.1) without a calculator. Directly computing the square root and then the reciprocal can be cumbersome. However, using our binomial expansion, we can approximate this value by substituting x = 0.1 into the polynomial. This gives us an approximate value of 1/2 + 0.1/16 + 3*(0.1)²/256, which is a sum of simple terms that can be easily calculated. This yields an approximation that is quite close to the true value, especially when x is small. The accuracy of the approximation improves as we include more terms from the binomial expansion, but even the first three terms often provide a reasonable estimate for small x. This technique is widely used in various fields, including physics and engineering, where approximations are frequently used to simplify complex calculations and models. Furthermore, binomial expansions play a crucial role in numerical analysis, where they are used to develop algorithms for approximating functions and solving equations. The ability to create accurate approximations is essential in many scientific and engineering applications, and binomial expansions provide a powerful tool for achieving this.

Conclusion

In summary, we've successfully found the first three terms of the binomial expansion of 1/√(4-x), which are 1/2, x/16, and 3x²/256. This exercise has not only demonstrated the application of the binomial theorem but also highlighted its practical use in approximating complex expressions. The binomial expansion is a fundamental tool in mathematics, with applications ranging from simplifying calculations to solving complex problems in various fields. Understanding and mastering this technique is crucial for anyone pursuing further studies in mathematics, science, or engineering. The process we followed, from rewriting the expression to applying the binomial theorem and simplifying the terms, is a standard approach that can be applied to a wide range of binomial expansion problems. The ability to manipulate expressions, recognize patterns, and apply formulas accurately are key skills that are developed through exercises like this. Moreover, the understanding of how these expansions can be used to approximate function values underscores the importance of mathematical tools in real-world applications. As you continue your mathematical journey, the concepts and techniques learned here will serve as a solid foundation for tackling more advanced topics and challenges. The power of the binomial theorem lies not only in its ability to expand expressions but also in its versatility and applicability in diverse mathematical and scientific contexts. By mastering this theorem, you gain a valuable tool that will enhance your problem-solving abilities and deepen your understanding of mathematics.