Problem 24 - Algebra and Trigonometry

Question

\(21-24\) Use the Binomial Theorem to expand the expression. $$ \left(2 A+B^{2}\right)^{4} $$

Step-by-Step Solution

Answer

The expanded form of \((2A+B^2)^4\) is \(16A^4 + 32A^3B^2 + 24A^2B^4 + 8AB^6 + B^8\).

Step 1: Understand the Binomial Theorem

The Binomial Theorem provides a way to expand expressions that are raised to a power. For any integers \(n\) and \(a\), \(b\), the formula is \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). Here, \(\binom{n}{k}\) represents the binomial coefficient calculated as \(\frac{n!}{k!(n-k)!}\).

Step 2: Identify Components in the Expression

In the expression \((2A+B^2)^4\), identify \(a = 2A\), \(b = B^2\), and \(n = 4\). We will use these values in the Binomial Theorem to expand the expression.

Step 3: Apply the Binomial Theorem

Now use \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\) to expand the expression:1. \(k = 0\): \(\binom{4}{0}(2A)^4(B^2)^0 = 1 \times 16A^4 = 16A^4\)2. \(k = 1\): \(\binom{4}{1}(2A)^3(B^2)^1 = 4 \times 8A^3 \times B^2 = 32A^3B^2\)3. \(k = 2\): \(\binom{4}{2}(2A)^2(B^2)^2 = 6 \times 4A^2 \times B^4 = 24A^2B^4\)4. \(k = 3\): \(\binom{4}{3}(2A)^1(B^2)^3 = 4 \times 2A \times B^6 = 8AB^6\)5. \(k = 4\): \(\binom{4}{4}(2A)^0(B^2)^4 = 1 \times B^8 = B^8\)

Step 4: Combine the Terms

Upon combining all the expanded terms from each step, the result is:\(16A^4 + 32A^3B^2 + 24A^2B^4 + 8AB^6 + B^8\).

Key Concepts

Expansion of Expressions

Expanding expressions often involves expressing a power of a binomial as a sum of terms. The Binomial Theorem is a powerful tool that helps simplify this process. It breaks down complex powers into manageable terms using a systematic approach. For any binomial - The formula is given by \[ (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \] - Here, the terms are expanded based on different powers of the components This means, for any two terms (like \(2A\) and \(B^2\) in our example), you can find each term's contribution to the final series. Expansion makes it easier to work with polynomials when solving equations or simplifying expressions. It turns a complex power into a series of additive terms, each involving products and powers of the original components.

Binomial Coefficients

Binomial coefficients are crucial components of the Binomial Theorem. They represent the number of ways to choose \(k\) items from a set of \(n\) items and are denoted by \( \binom{n}{k} \). The formula to calculate a binomial coefficient is - Given as \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] - Where \(n!\) (n factorial) is the product of all positive integers up to \(n\) These coefficients appear in each term of the binomial expansion and dictate the weight or proportion of each contribution to the total expansion. For example, during the expansion of \((2A + B^2)^4\), coefficients like \(\binom{4}{0} = 1\) and \(\binom{4}{1} = 4\) scale the terms such as \(16A^4\) and \(32A^3B^2\), ensuring the correct additive combination in the polynomial.

Combinatorics

Combinatorics is the branch of mathematics dealing with combinations, permutations, and the count of different possible arrangements. In the context of the Binomial Theorem, combinatorics explains the distribution of binomial coefficients. It provides insights into why and how each term in a binomial expansion is structured.The arrangement and repetition principles outlined by combinatorics help us understand the rationale behind each term's construction in the expansion:- Every term corresponds to a different combination of powers of \(a\) and \(b\) - The exponent \(k\) indicates how the power or component of \(b\) is utilized Understanding these principles allows us to delve deeper into polynomials and provides a mathematical foundation for more complex topic areas like probability and statistics.

Polynomials

Polynomials are expressions made up of variables and coefficients involving operations like addition, subtraction, and non-negative integer exponents of variables.- In the expansion of \((2A + B^2)^4\), we combine terms to form a polynomial The expanded form \(16A^4 + 32A^3B^2 + 24A^2B^4 + 8AB^6 + B^8\) is a polynomial consisting of different powers of \(A\) and \(B\)Polynomials are significant in mathematics because they can express a wide range of complex relationships and functions in a simplified form. They are used in algebra, calculus, and numerous applications in science and engineering, making it essential to grasp how to expand and manipulate them efficiently using tools like the Binomial Theorem.