Problem 24 - Precalculus: Mathematical for Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the terms

In the expression \((2A + B^2)^4\), identify the first term as \(2A\) and the second term as \(B^2\). The power is 4.

Step 2: Write the Binomial Theorem formula

The Binomial Theorem states that \((x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\). In this case, \(x = 2A\), \(y = B^2\), and \(n = 4\).

Step 3: Calculate binomial coefficients

Calculate the binomial coefficients \(\binom{4}{0}, \binom{4}{1}, \binom{4}{2}, \binom{4}{3},\) and \(\binom{4}{4}\). These are 1, 4, 6, 4, and 1, respectively.

Step 4: Write terms using Binomial Theorem

Expand the expression using the calculated coefficients and the formula: \[ (2A)^4(B^2)^0 + \binom{4}{1}(2A)^3(B^2)^1 + \binom{4}{2}(2A)^2(B^2)^2 + \binom{4}{3}(2A)^1(B^2)^3 + \binom{4}{4}(2A)^0(B^2)^4 \]

Step 5: Simplify each term

Calculate each term individually: 1. \((2A)^4(B^2)^0 = 16A^4\)2. \(4 \times (2A)^3(B^2)^1 = 32A^3B^2\)3. \(6 \times (2A)^2(B^2)^2 = 24A^2B^4\)4. \(4 \times (2A)^1(B^2)^3 = 8AB^6\)5. \((2A)^0(B^2)^4 = B^8\)

Step 6: Combine all terms

Put all simplified terms together: \(16A^4 + 32A^3B^2 + 24A^2B^4 + 8AB^6 + B^8\).

Key Concepts

Binomial expansion

The binomial expansion is a way to write expressions of the form \((x + y)^n\) in an expanded form. This is particularly useful in simplifying algebraic expressions without multiplying them out directly.
By using the Binomial Theorem, we can expand binomials raised to any power by applying a well-defined formula. The theorem states that:

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

Here, \(n\) is the power to which the binomial is raised, and \(x\) and \(y\) are the terms inside the binomial. The symbol \(\binom{n}{k}\) represents the binomial coefficient.
The expression in our example \((2A + B^2)^4\) is expanded using the Binomial Theorem, which makes it easier to simplify without manually multiplying it four times. By strategically applying this theorem, each term in the power series can be derived, showing its practical usefulness in solving algebraic equations.

Combinatorial coefficients

Combinatorial coefficients, commonly known as binomial coefficients, play a crucial role in the binomial expansion. They appear in the polynomial expansion of a binomial expression and are denoted by \(\binom{n}{k}\).
These coefficients are calculated using the formula:

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

Where \(n!\) (n factorial) is the product of all positive integers up to \(n\), and \(k\) is the index of the term in the expansion. In the given exercise, the binomial coefficients for the power 4 are 1, 4, 6, 4, and 1.
These coefficients determine how each term in the binomial expansion is weighted. For instance, in the expression \((2A + B^2)^4\), the coefficient 6 comes from \(\binom{4}{2}\) and affects the term \((2A)^2(B^2)^2\). Understanding these coefficients allows for precise calculation and simplification of polynomial expressions.

Polynomial expressions

Polynomial expressions are algebraic expressions comprised of variables and coefficients, constructed using the operations of addition, subtraction, and non-negative integer exponents of variables.
They form the basis of many algebraic operations and are fundamental in calculus, algebra, and beyond. In the context of binomial expansion, we frequently encounter polynomial expressions as the result of expanding a binomial raised to a power.
Let's look at our case: the expanded expression \((2A + B^2)^4\) results in the polynomial \(16A^4 + 32A^3B^2 + 24A^2B^4 + 8AB^6 + B^8\). Each term is itself a simplified polynomial expression, each with its own variable terms and coefficients.
Polynomials can be classified based on their degree (the highest power of the variable) or the number of terms. For instance, the resulting expanded expression is a 5-term polynomial because it contains five distinct terms, making it a quintic polynomial due to the highest combined degree of 8 in \(B^8\).