Finding The Coefficient Of X In (1 + 4/x)^2 (1 + X/4)^8
In the realm of algebra, expanding expressions and identifying coefficients of specific terms is a fundamental skill. This article delves into the process of finding the coefficient of x in the expansion of the expression (1 + 4/x)^2 (1 + x/4)^8. We will embark on a step-by-step journey, employing algebraic manipulation and the binomial theorem to unravel this mathematical puzzle. Our goal is to provide a comprehensive and insightful explanation, making the process accessible to both seasoned mathematicians and those new to the field.
Understanding the Binomial Theorem: A Foundation for Expansion
The cornerstone of our approach is the binomial theorem, a powerful tool for expanding expressions of the form (a + b)^n, where n is a non-negative integer. The binomial theorem provides a systematic way to determine the coefficients and terms in the expansion. At its heart, the theorem states:
(a + b)^n = Σ (n choose k) * a^(n-k) * b^k
where the summation (Σ) is taken over all integer values of k from 0 to n, and (n choose k) represents the binomial coefficient, often read as "n choose k". This coefficient quantifies the number of ways to choose k objects from a set of n distinct objects and is mathematically expressed as:
(n choose k) = n! / (k! * (n-k)!)
where "!" denotes the factorial function (e.g., 5! = 5 * 4 * 3 * 2 * 1). The binomial theorem serves as our guiding principle, allowing us to systematically expand the given expression and isolate the term containing x. By understanding its mechanics, we can effectively tackle the problem at hand.
Expanding (1 + 4/x)^2: A Crucial First Step
Before we can tackle the entire expression, we must first expand the term (1 + 4/x)^2. This initial expansion is crucial as it lays the foundation for the subsequent steps. Expanding this term, we get:
(1 + 4/x)^2 = 1^2 + 2 * 1 * (4/x) + (4/x)^2 = 1 + 8/x + 16/x^2
This expansion reveals three distinct terms: a constant term (1), a term with x in the denominator (8/x), and a term with x squared in the denominator (16/x^2). Each of these terms will interact with the expansion of the second part of the expression, (1 + x/4)^8, and contribute to the final coefficient of x. It is important to note that only certain terms from the expansion of (1 + x/4)^8, when multiplied with these terms, will result in a term with x to the power of 1. Specifically, the constant term (1) will need to be multiplied by the x term from the expansion of (1 + x/4)^8, the 8/x term will need to be multiplied by the constant term, and the 16/x^2 term will need to be multiplied by the x^3 term. This careful consideration of term interactions is key to correctly identifying the final coefficient.
Expanding (1 + x/4)^8: Unveiling the Potential for x Terms
Now, let's turn our attention to the second part of the expression: (1 + x/4)^8. To expand this, we invoke the binomial theorem once more. Applying the theorem, we have:
(1 + x/4)^8 = Σ (8 choose k) * 1^(8-k) * (x/4)^k
where k ranges from 0 to 8. This expansion will yield nine terms, each with a different power of x. However, we are not interested in all of these terms. Our focus is on identifying the terms that, when multiplied by the terms from the expansion of (1 + 4/x)^2, will result in a term with x. As previously discussed, these are the constant term (k=0), the x term (k=1), and the x^2 term (k=2). By isolating these terms, we can simplify our calculations and directly target the coefficient of x. The binomial theorem empowers us to systematically dissect this expansion and pinpoint the relevant terms.
Identifying Key Terms: The Constant, x, and x² Contributions
To find the coefficient of x, we need to identify the terms in the expansion of (1 + x/4)^8 that, when multiplied by the terms in the expansion of (1 + 4/x)^2, will yield a term with x. As established earlier, these are the constant term, the x term, and the x² term. Let's calculate these terms individually:
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Constant Term (k=0): (8 choose 0) * 1^(8-0) * (x/4)^0 = 1 * 1 * 1 = 1
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x Term (k=1): (8 choose 1) * 1^(8-1) * (x/4)^1 = 8 * 1 * (x/4) = 2x
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x² Term (k=2): (8 choose 2) * 1^(8-2) * (x/4)^2 = 28 * 1 * (x^2/16) = (7/4)x²
These three terms are the crucial building blocks for finding the coefficient of x. By carefully considering how they interact with the terms from the first expansion, we can isolate the contributions to the final coefficient. This meticulous approach ensures that we account for all relevant terms and avoid any oversights in our calculation.
The Final Calculation: Summing the Contributions to the x Coefficient
Now, we have all the pieces of the puzzle. We have expanded both parts of the expression and identified the key terms that contribute to the coefficient of x. The final step is to multiply the appropriate terms from each expansion and sum their contributions. Let's break it down:
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Multiplying the constant term from (1 + 4/x)^2 (1) by the x term from (1 + x/4)^8 (2x): 1 * 2x = 2x
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Multiplying the 8/x term from (1 + 4/x)^2 by the constant term from (1 + x/4)^8 (1): (8/x) * 0 (no x^-1 term) = 0
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Multiplying the 16/x^2 term from (1 + 4/x)^2 by the x² term from (1 + x/4)^8 ((7/4)x²): (16/x^2) * 0 (no x^2 term that cancels) = 0
Therefore, adding the x term gives us:
2x + 0 + 0= 2x
Thus, the coefficient of x in the expansion of (1 + 4/x)^2 (1 + x/4)^8 is 2. This final calculation represents the culmination of our step-by-step analysis, demonstrating the power of algebraic manipulation and the binomial theorem in solving complex mathematical problems.
Conclusion: A Journey Through Expansion and Coefficient Identification
In this article, we have successfully navigated the expansion of (1 + 4/x)^2 (1 + x/4)^8 and determined the coefficient of x. Our journey began with an understanding of the binomial theorem, a fundamental tool for expanding expressions of the form (a + b)^n. We then meticulously expanded each part of the expression, (1 + 4/x)^2 and (1 + x/4)^8, paying close attention to the terms that would contribute to the coefficient of x. By identifying the constant term, the x term, and the x² term in the expansion of (1 + x/4)^8, we were able to isolate the relevant contributions. Finally, we multiplied the appropriate terms and summed their coefficients, arriving at the final answer: 2. This process underscores the importance of a systematic approach to algebraic problems. By breaking down a complex problem into smaller, manageable steps, we can effectively leverage mathematical tools and techniques to arrive at a solution. The binomial theorem, in particular, provides a powerful framework for expanding expressions and extracting specific coefficients. This exploration not only demonstrates the practical application of these concepts but also reinforces the beauty and elegance of mathematical reasoning.