Binomial Expansion Finding First Three Non-Zero Terms

Introduction to Binomial Expansion

In this article, we will delve into the fascinating world of binomial expansions. Specifically, we will focus on finding the first three non-zero terms in the expansion of the expression ${\left(2-\frac{x^2}{4}\right)^6}$ in ascending powers of ${x}$. This problem requires a solid understanding of the binomial theorem and its applications. The binomial theorem provides a powerful tool for expanding expressions of the form ${(a + b)^n}$, where ${n}$ is a non-negative integer. It is widely used in various fields of mathematics, including algebra, calculus, and combinatorics. Mastery of binomial expansion is crucial for solving a wide range of mathematical problems.

Before diving into the specifics of our problem, let's briefly review the binomial theorem. The theorem states that for any non-negative integer ${n}$ and any real numbers ${a}$ and ${b}$, the expansion of ${(a + b)^n}$ can be written as:

$$(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$$

Where ${\binom{n}{k}}$ represents the binomial coefficient, also known as the combination, and is calculated as:

$$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$

Here, ${n!}$ denotes the factorial of ${n}$, which is the product of all positive integers up to ${n}$. Understanding these basics is critical for effectively tackling binomial expansion problems. In the context of our given expression, ${a = 2}$, ${b = -\frac{x^2}{4}}$, and ${n = 6}$. Our goal is to find the first three terms in the expansion, which correspond to the terms with the lowest powers of ${x}$.

Applying the Binomial Theorem

To apply the binomial theorem to our specific expression, ${\left(2-\frac{x^2}{4}\right)^6}$, we will substitute the values of ${a}$, ${b}$, and ${n}$ into the binomial theorem formula. The binomial expansion will yield a series of terms, each with a different power of ${x}$. We are interested in finding the first three non-zero terms, which will correspond to the terms with the lowest powers of ${x}$. The general term in the binomial expansion is given by:

$$T_{k+1} = \binom{n}{k} a^{n-k} b^k$$

In our case, this translates to:

$$T_{k+1} = \binom{6}{k} (2)^{6-k} \left(-\frac{x^2}{4}\right)^k$$

We need to find the terms for ${k = 0}$, ${k = 1}$, and ${k = 2}$ to obtain the first three terms. Let's calculate these terms one by one. For ${k = 0}$, we have:

$$T_1 = \binom{6}{0} (2)^{6-0} \left(-\frac{x^2}{4}\right)^0 = 1 \cdot 2^6 \cdot 1 = 64$$

For ${k = 1}$, we have:

$$T_2 = \binom{6}{1} (2)^{6-1} \left(-\frac{x^2}{4}\right)^1 = 6 \cdot 2^5 \cdot \left(-\frac{x^2}{4}\right) = 6 \cdot 32 \cdot \left(-\frac{x^2}{4}\right) = -48x^2$$

For ${k = 2}$, we have:

$$T_3 = \binom{6}{2} (2)^{6-2} \left(-\frac{x^2}{4}\right)^2 = \frac{6!}{2!4!} \cdot 2^4 \cdot \left(\frac{x^4}{16}\right) = 15 \cdot 16 \cdot \frac{x^4}{16} = 15x^4$$

Thus, the first three non-zero terms in the expansion are 64, -48x^2, and 15x^4. These terms represent the constant term, the ${x^2}$ term, and the ${x^4}$ term, respectively. The binomial coefficients play a critical role in determining the magnitude of each term in the expansion. The pattern observed in the exponents of ${x}$ is also a key aspect of binomial expansions, where the exponents increase by a fixed amount with each subsequent term.

Simplifying the Terms

In this section, we will simplify each term obtained from the binomial expansion. This involves performing the necessary arithmetic operations to express each term in its simplest form. We have already calculated the first three terms as:

• Term 1: 64
• Term 2: -48x^2
• Term 3: 15x^4

The first term, 64, is already a constant and does not require further simplification. It is the leading term in the expansion and represents the value of the expression when ${x = 0}$. The second term, -48x^2, is a quadratic term. It is already in its simplest form, with the coefficient -48 multiplying the squared variable ${x^2}$. This term contributes to the overall shape of the function represented by the binomial expansion, particularly its curvature near ${x = 0}$. The negative sign indicates that this term will decrease the value of the expression as ${x}$ moves away from zero in either direction. The third term, 15x^4, is a quartic term. Like the second term, it is already in its simplest form, with the coefficient 15 multiplying the fourth power of ${x}$. This term has a higher power of ${x}$, which means it will become more significant as ${x}$ increases in magnitude. The positive coefficient indicates that this term will increase the value of the expression as ${x}$ moves away from zero.

The simplified terms provide a clear representation of the initial behavior of the binomial expansion. The constant term gives the value of the expression at ${x = 0}$, while the higher-order terms describe how the expression changes as ${x}$ varies. Simplifying these terms allows us to easily identify their contributions to the overall expansion and facilitates further analysis, such as finding approximations or evaluating the expression for specific values of ${x}$. The process of simplification also ensures that the expansion is presented in a concise and understandable manner, which is crucial for effective communication of mathematical results.

Ascending Powers of x

Arranging the terms in ascending powers of ${x}$ is a crucial step in expressing the binomial expansion in a standard form. This arrangement allows us to easily identify the dominant terms for small values of ${x}$ and provides insights into the behavior of the function near ${x = 0}$. The terms we found earlier are 64, -48x^2, and 15x^4. These terms are already in ascending order of powers of ${x}$, since the powers are 0, 2, and 4, respectively. Therefore, the first three non-zero terms in ascending powers of ${x}$ are:

$$64 - 48x^2 + 15x^4$$

This arrangement clearly shows the increasing order of the powers of ${x}$. The term with the lowest power of ${x}$ (the constant term) comes first, followed by the term with the next lowest power (the ${x^2}$ term), and so on. This is a common convention in mathematical notation, as it allows for a straightforward interpretation of the expression's behavior. In the context of this binomial expansion, the ascending power arrangement highlights the relative importance of each term for small values of ${x}$. When ${x}$ is close to zero, the constant term 64 dominates the expression, as the other terms involving powers of ${x}$ become very small. As ${x}$ increases, the terms with higher powers of ${x}$ start to contribute more significantly to the overall value of the expression.

The ascending power arrangement is particularly useful when we want to approximate the value of the expression for small values of ${x}$. By considering only the first few terms in the expansion, we can obtain a good approximation of the expression's value. For instance, if ${x}$ is very small, the first two terms (64 and -48x^2) might provide a sufficiently accurate approximation. This technique is widely used in various fields, such as physics and engineering, where it is often necessary to simplify complex expressions for practical calculations. Understanding the concept of ascending powers of ${x}$ is therefore essential for applying binomial expansions effectively in different contexts.

Conclusion

In conclusion, we have successfully found the first three non-zero terms in the expansion of ${\left(2-\frac{x^2}{4}\right)^6}$ in ascending powers of ${x}$. By applying the binomial theorem and simplifying each term, we obtained the expansion:

$$64 - 48x^2 + 15x^4$$

This process involved several key steps, including understanding and applying the binomial theorem, calculating binomial coefficients, simplifying algebraic expressions, and arranging terms in ascending order of powers of ${x}$. Each of these steps is fundamental to working with binomial expansions and requires a solid foundation in algebraic manipulation and mathematical reasoning. The binomial theorem is a powerful tool that allows us to expand expressions of the form ${(a + b)^n}$ into a sum of terms, each with a specific coefficient and power of the variables. The ability to apply this theorem accurately is crucial for solving a wide range of problems in mathematics, physics, and engineering. The calculated binomial coefficients determine the magnitude of each term in the expansion, while the powers of the variables indicate how the terms change as the variables vary.

Simplifying the terms obtained from the binomial expansion is an important step in presenting the results in a clear and concise manner. This involves performing any necessary arithmetic operations and combining like terms to express each term in its simplest form. Arranging the terms in ascending powers of ${x}$ provides a standard way of representing the expansion, which allows for easy identification of the dominant terms for small values of ${x}$. This arrangement is particularly useful when we want to approximate the value of the expression for small values of ${x}$, as we can often obtain a good approximation by considering only the first few terms in the expansion. The knowledge and skills gained from this exercise can be applied to more complex problems involving binomial expansions and other related mathematical concepts. Mastering these techniques is essential for anyone pursuing studies in mathematics, science, or engineering.