Binomial Expansion Problems Solutions And Examples

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Introduction to Binomial Expansion

In the realm of mathematics, binomial expansion stands as a cornerstone concept, particularly within algebra and calculus. It provides a systematic method for expanding expressions of the form $(x + y)^n$, where 'n' is a non-negative integer. This expansion is not merely a mathematical curiosity; it has profound applications across various fields, including probability, statistics, and even computer science. Understanding binomial expansion is crucial for solving a wide range of problems, from simplifying complex algebraic expressions to approximating numerical values. This exploration delves into the intricacies of binomial expansion, focusing on solving problems that involve finding specific coefficients and terms within the expansion. Our primary focus will be on deciphering the values of 'a' and 'b' in the expansion of $(2-x)(1+ax)^6$, a problem that elegantly combines binomial theorem principles with algebraic manipulation. Further, we will embark on a journey to identify the initial three terms of a given binomial expansion, a task that requires a firm grasp of the binomial theorem and its applications. As we unravel these problems, we will not only enhance our problem-solving skills but also gain a deeper appreciation for the elegance and utility of binomial expansion.

Understanding the Binomial Theorem

The binomial theorem is the bedrock of binomial expansion. It provides a formula for expanding expressions of the form $(x + y)^n$, where 'n' is a non-negative integer. The theorem states that:

$(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k$

Here, $\binom{n}{k}$ represents the binomial coefficient, often read as "n choose k," and is calculated as:

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

Where '!' denotes the factorial function. The binomial coefficients hold a treasure trove of information about the expansion, dictating the numerical factors that multiply each term. These coefficients possess symmetrical properties, which can simplify calculations. For instance, $\binom{n}{k}$ is always equal to $\binom{n}{n-k}$. This symmetry not only reduces the computational burden but also provides a deeper understanding of the structure of binomial expansions. Moreover, the binomial coefficients can be visually represented in Pascal's Triangle, a triangular array of numbers where each number is the sum of the two numbers directly above it. Pascal's Triangle offers a quick way to determine binomial coefficients for small values of 'n', making it a valuable tool for manual calculations and conceptual understanding. The binomial theorem's significance extends beyond its formulaic representation; it embodies a fundamental principle of combinatorial mathematics, linking algebra with combinatorics in a profound way.

Applying the Binomial Theorem to Specific Problems

To effectively apply the binomial theorem, let's consider the problem of finding the expansion of $(1 + ax)^6$. Using the binomial theorem, we can expand this expression as follows:

$(1 + ax)^6 = \binom{6}{0}1^6(ax)^0 + \binom{6}{1}1^5(ax)^1 + \binom{6}{2}1^4(ax)^2 + \cdots$

Calculating the binomial coefficients, we get:

$\binom{6}{0} = 1$ $\binom{6}{1} = 6$ $\binom{6}{2} = 15$

Substituting these values back into the expansion, we obtain:

$(1 + ax)^6 = 1 + 6ax + 15a^2x^2 + \cdots$

This expansion forms the basis for solving more complex problems involving binomial expansions. When faced with problems that require finding specific terms or coefficients, it's crucial to first identify the general term in the expansion. The general term is a formula that represents any term in the expansion, allowing us to pinpoint the term corresponding to a specific power of 'x'. By setting up equations based on the given information, such as the coefficients of certain terms, we can solve for unknown variables. This methodical approach transforms the often daunting task of expanding binomials into a manageable and insightful exercise. Moreover, understanding the pattern of coefficients and exponents in the binomial expansion not only aids in problem-solving but also enhances our appreciation for the elegant structure inherent in mathematical expressions.

Solving for 'a' and 'b' in the Expansion of $(2-x)(1+ax)^6$

Setting up the Equation

The problem states that the expansion of $(2-x)(1+ax)^6$ as far as the term in $x^2$ is $1 + bx^2 + \cdots$. Our goal is to find the values of 'a' and 'b'. To achieve this, we first need to expand $(1+ax)^6$ up to the $x^2$ term, which we already accomplished in the previous section. Now, we multiply this expansion by $(2-x)$:

$(2-x)(1+ax)^6 = (2-x)(1 + 6ax + 15a^2x^2 + \cdots)$

Expanding this product, we focus on collecting terms up to $x^2$:

$2(1 + 6ax + 15a^2x^2) - x(1 + 6ax + 15a^2x^2) + \cdots$

This step is crucial as it sets the stage for comparing coefficients and forming equations. By meticulously multiplying and combining terms, we ensure that no relevant terms are overlooked. The process not only involves algebraic manipulation but also requires a keen eye for detail, ensuring that each term is correctly accounted for. Moreover, by focusing on terms up to $x^2$, we streamline the problem-solving process, allowing us to directly address the unknowns 'a' and 'b' without getting bogged down in higher-order terms. This strategic approach exemplifies the power of problem simplification, a valuable skill in mathematics and beyond.

Solving for 'a'

Now, let's expand and collect like terms:

$2 + 12ax + 30a^2x^2 - x - 6ax^2 + \cdots$

Combining the terms, we get:

$2 + (12a - 1)x + (30a^2 - 6a)x^2 + \cdots$

We are given that the expansion is of the form $1 + bx^2 + \cdots$. This implies that the coefficient of the $x$ term must be zero. Therefore:

$12a - 1 = 0$

Solving for 'a', we find:

$a = \frac{1}{12}$

This step demonstrates the power of equating coefficients, a fundamental technique in algebra. By recognizing that the coefficient of 'x' must be zero, we transform the problem into a simple algebraic equation, which can be easily solved. This approach highlights the importance of connecting given information with the structure of the expansion. Moreover, the solution for 'a' not only provides a numerical value but also serves as a critical piece of the puzzle, paving the way for determining the value of 'b'. The methodical nature of this process underscores the beauty of mathematical reasoning, where each step builds upon the previous one, leading to a coherent and satisfying solution.

Solving for 'b'

Now that we have the value of 'a', we can find 'b' by equating the coefficients of the $x^2$ terms:

$b = 30a^2 - 6a$

Substituting $a = \frac{1}{12}$, we get:

$b = 30(\frac{1}{12})^2 - 6(\frac{1}{12})$ $b = 30(\frac{1}{144}) - \frac{1}{2}$ $b = \frac{30}{144} - \frac{1}{2}$ $b = \frac{5}{24} - \frac{1}{2}$ $b = \frac{5 - 12}{24}$ $b = -\frac{7}{24}$

Thus, the value of $b$ is $-\frac{7}{24}$. This final calculation showcases the importance of accurate substitution and arithmetic. By carefully plugging in the value of 'a', we unravel the value of 'b', completing the solution to the problem. The process underscores the interconnectedness of mathematical concepts, where the solution to one variable directly influences the solution to another. Moreover, the final answer not only provides a numerical value but also reaffirms the power of algebraic manipulation and problem-solving strategies. This journey, from setting up the equation to finding the final answer, exemplifies the satisfaction that comes from tackling mathematical challenges with a clear and methodical approach.

Finding the First Three Terms of a Binomial Expansion

Applying the Binomial Theorem

The second part of the problem requires us to find the first three terms of a binomial expansion. Let's consider a general binomial expression $(A + B)^n$. The first three terms of its expansion can be found using the binomial theorem:

Term 1: $\binom{n}{0}A^nB^0$ Term 2: $\binom{n}{1}A^{n-1}B^1$ Term 3: $\binom{n}{2}A^{n-2}B^2$

This framework provides a systematic way to approach any binomial expansion. By understanding the pattern of binomial coefficients and exponents, we can efficiently determine the initial terms of the expansion. The first term always involves the binomial coefficient $\binom{n}{0}$, which is equal to 1, simplifying the calculation. The second term introduces the binomial coefficient $\binom{n}{1}$, which is equal to 'n', providing a direct link between the exponent of the binomial and the coefficient of the second term. The third term involves $\binom{n}{2}$, which can be calculated using the formula $\frac{n(n-1)}{2}$, adding another layer of complexity to the calculation. However, by mastering these patterns, we can confidently navigate the world of binomial expansions and extract valuable information from them.

Example and Calculation

For instance, if we have the expression $(2 + 3x)^5$, we can find the first three terms as follows:

Term 1: $\binom{5}{0}(2)^5(3x)^0 = 1 imes 32 imes 1 = 32$ Term 2: $\binom{5}{1}(2)^4(3x)^1 = 5 imes 16 imes 3x = 240x$ Term 3: $\binom{5}{2}(2)^3(3x)^2 = 10 imes 8 imes 9x^2 = 720x^2$

Thus, the first three terms of the expansion are $32 + 240x + 720x^2$. This example showcases the practical application of the binomial theorem in finding specific terms within an expansion. By systematically applying the formula and carefully calculating the binomial coefficients and exponents, we can efficiently determine the desired terms. The process not only demonstrates the power of the binomial theorem but also highlights the importance of precision in mathematical calculations. Moreover, by understanding the underlying patterns and principles, we can confidently tackle a wide range of binomial expansion problems, from simple cases to more complex scenarios.

Conclusion

In conclusion, the problem involving the expansion of $(2-x)(1+ax)^6$ and finding its first three terms encapsulates the core principles of binomial expansion. We successfully determined the values of $a$ and $b$ by applying the binomial theorem, equating coefficients, and solving the resulting equations. Furthermore, we explored the method for finding the initial terms of a binomial expansion, reinforcing the application of the binomial theorem in diverse scenarios. These exercises not only enhance our algebraic skills but also deepen our understanding of the binomial theorem's versatility and its significance in mathematics. The ability to expand binomial expressions and extract specific information from them is a valuable asset in various mathematical contexts, from calculus to probability. By mastering these concepts, we empower ourselves to tackle complex problems with confidence and precision, solidifying our foundation in the world of mathematics.