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

Step 1: Identify the Terms

In the expression \( (2A + B^2)^4 \), the terms are \( a = 2A \) and \( b = B^2 \), and the exponent is 4.

Step 2: Recall the Binomial Theorem

The Binomial Theorem states that \( (x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k \). We will use this formula to expand \( (2A + B^2)^4 \).

Step 3: Apply the Binomial Theorem

For \( n = 4 \), substitute \( a = 2A \) and \( b = B^2 \) into the Binomial Theorem: \[(2A + B^2)^4 = \sum_{k=0}^{4} \binom{4}{k} (2A)^{4-k} (B^2)^k\]

Step 4: Calculate Each Term in the Expansion

Compute each term for \( k = 0 \) to \( k = 4 \):- For \( k = 0 \): \( \binom{4}{0} (2A)^4 (B^2)^0 = 1 \times 16A^4 = 16A^4 \)- For \( k = 1 \): \( \binom{4}{1} (2A)^3 (B^2)^1 = 4 \times 8A^3 \times B^2 = 32A^3B^2 \)- For \( k = 2 \): \( \binom{4}{2} (2A)^2 (B^2)^2 = 6 \times 4A^2 \times B^4 = 24A^2B^4 \)- For \( k = 3 \): \( \binom{4}{3} (2A)^1 (B^2)^3 = 4 \times 2A \times B^6 = 8AB^6 \)- For \( k = 4 \): \( \binom{4}{4} (2A)^0 (B^2)^4 = 1 \times B^8 = B^8 \)

Step 5: Write the Expanded Expression

Combine all the calculated terms to get the expanded expression:\[ 16A^4 + 32A^3B^2 + 24A^2B^4 + 8AB^6 + B^8 \]

Key Concepts

Polynomial Expansion

A polynomial expansion involves expressing a polynomial raised to a power as a sum of terms. Each term involves the constituents of the original polynomial, but with altered exponents and coefficients. In this exercise, we focus on expanding the expression \((2A + B^2)^4\) using the Binomial Theorem. Expanding polynomials is vital since it allows us to convert a factored expression into a form that is easier to integrate, differentiate, or evaluate at particular values.
When using the Binomial Theorem for polynomial expansion, each term in the expansion takes the form \(\binom{n}{k} a^{n-k} b^k\), where \(a\) and \(b\) are the terms being expanded, \(n\) is the power, and \(k\) varies from 0 to \(n\).
This technique significantly simplifies the computation of higher powers of polynomials by systematically calculating each term rather than manually multiplying out the polynomial.

Algebraic Expressions

Algebraic expressions form the backbone of algebra, acting as phrases of mathematical language composed of numbers, variables, and operation signs. In the expression \((2A + B^2)\), \(2A\) and \(B^2\) are algebraic terms.
An expression becomes more complex when it involves multiple operations, like addition, subtraction, multiplication, and division of these terms. The aim generally is to simplify these expressions or manipulate them to solve equations.
The complexity in our exercise arises from the exponential form \((2A + B^2)^4\). Using the Binomial Theorem in algebraic expressions facilitates simplification and expansion, leading to the clear and explicit representation \(16A^4 + 32A^3B^2 + 24A^2B^4 + 8AB^6 + B^8\).

Combinatorics

Combinatorics plays a crucial role in the expansion of a binomial expression, particularly through the use of binomial coefficients. These coefficients, denoted as \(\binom{n}{k}\), represent the number of ways to choose \(k\) elements from \(n\) elements, also known as combinations.
When using the Binomial Theorem, these coefficients are paramount as they define the weight of each term in the expansion. For instance, in our example, the coefficients for \((2A + B^2)^4\) were calculated as follows:

• \(\binom{4}{0} = 1\)
• \(\binom{4}{1} = 4\)
• \(\binom{4}{2} = 6\)
• \(\binom{4}{3} = 4\)
• \(\binom{4}{4} = 1\)

These coefficients multiply the algebraic terms in the expanded form, underscoring the direct relationship between combinatorics and polynomial expansion.

Exponentiation

Exponentiation is the mathematical operation involving the raising of a number or expression to a power. In the context of the binomial expression \((2A + B^2)^4\), exponentiation is crucial in calculating each term.
Each part of the expression \((2A)^{4-k}\) and \((B^2)^k\) involves raising \(2A\) and \(B^2\) to various powers, based on the term number \(k\) as dictated by the Binomial Theorem formula.
Understanding how to manipulate exponents is key, as it governs the progression of each term's degree from the initial polynomial to the fully expanded equation. This systematic approach simplifies solving complex expressions like \((2A + B^2)^4\), ultimately aiding in further algebraic analysis and problem-solving.