Problem 28 - Precalculus: Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The expansion is \(16A^4 + 32A^3 B^2 + 24A^2 B^4 + 8A B^6 + B^8\).

Step 1: Identify the binomial components

The given expression is \((2A + B^2)^4\). Here, the first term \(a = 2A\) and the second term \(b = B^2\). The exponent \(n = 4\).

Step 2: Recall the Binomial Theorem

The Binomial Theorem states: \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). Apply this theorem to the given expression using \(a = 2A\), \(b = B^2\), and \(n = 4\).

Step 3: Calculate the binomial coefficients

Use the binomial coefficient formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\) to determine the coefficients for each term. For \(n = 4\):\- \(\binom{4}{0} = 1\)\- \(\binom{4}{1} = 4\)\- \(\binom{4}{2} = 6\)\- \(\binom{4}{3} = 4\)\- \(\binom{4}{4} = 1\)

Step 4: Expand the expression term by term

Using the Binomial Theorem, expand \((2A + B^2)^4 = \sum_{k=0}^{4} \binom{4}{k} (2A)^{4-k} (B^2)^k\). Calculate each term as follows:\- For \(k = 0\): \(\binom{4}{0} (2A)^4 (B^2)^0 = 1 \cdot 16A^4 = 16A^4\)\- For \(k = 1\): \(\binom{4}{1} (2A)^3 (B^2)^1 = 4 \cdot 8A^3 B^2 = 32A^3 B^2\)\- For \(k = 2\): \(\binom{4}{2} (2A)^2 (B^2)^2 = 6 \cdot 4A^2 B^4 = 24A^2 B^4\)\- For \(k = 3\): \(\binom{4}{3} (2A)^1 (B^2)^3 = 4 \cdot 2A B^6 = 8A B^6\)\- For \(k = 4\): \(\binom{4}{4} (2A)^0 (B^2)^4 = 1 \cdot B^8 = B^8\)

Step 5: Sum the terms to obtain the final expansion

Add up all the terms calculated in Step 4: \(16A^4 + 32A^3 B^2 + 24A^2 B^4 + 8A B^6 + B^8\). This is the expanded form of the expression.

Key Concepts

Binomial Expansion

The technique of binomial expansion is a powerful tool in algebra, especially when it comes to expanding expressions raised to a power. The Binomial Theorem provides a formula for expanding expressions of the form \((a+b)^n\). This theorem saves time and effort by systematically determining each term in the expansion without multiplying everything manually.
To apply this theorem, you identify two components in the expression: the first term \(a\) and the second term \(b\), along with the exponent \(n\). For example, in the expression \((2A + B^2)^4\), \(a\) is \(2A\) and \(b\) is \(B^2\). The theorem then constructs each term in the expansion based on a combination of both terms raised to appropriate powers.
The terms are calculated by multiplying a binomial coefficient with all possible combinations of the powers of \(a\) and \(b\) that sum to \(n\). This technique is very beneficial in simplifying problems, especially as the exponent increases.

Binomial Coefficient

A key component in binomial expansion is the binomial coefficient, which appears in each term of the expanded expression. Binomial coefficients determine the weight of each term and are represented as \(\binom{n}{k}\), often read as "\(n\) choose \(k\)". These coefficients indicate how many ways you can choose \(k\) items from a set of \(n\).
To compute a binomial coefficient, use the formula: \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\). The exclamation mark denotes a factorial, meaning a product of all positive integers up to that number. For example, \(\binom{4}{2}\) is calculated as \(\frac{4!}{2!(4-2)!} = \frac{4 \times 3 \times 2 \times 1}{2 \times 1 \times 2 \times 1} = 6\).
These coefficients give balance in the polynomial expansion by ensuring each power combination appears the correct number of times.

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. It is foundational for understanding concepts like binomial expansion because it allows for generalized, abstract thinking.
In binomial expansion, algebra helps us manage operations with variables as though they were numerical quantities. It provides us with the rules and structures needed to solve equations, transform expressions, and, in this case, expand binomials.

• Algebraic expressions can include coefficients, variables, and exponents.
• They follow specific laws and patterns, like the distributive, associative, and commutative properties.

Understanding algebra is crucial because it creates a foundation for advancing into more complex areas of mathematics, enabling the effective application of the binomial theorem and other crucial theorems in solving real-world problems.