Problem 28 - Algebra 2

Question

Expand each power. $$ (2 a+b)^{6} $$

Step-by-Step Solution

Answer

Expanded expression: \(64a^6 + 192a^5b + 240a^4b^2 + 160a^3b^3 + 60a^2b^4 + 12ab^5 + b^6\).

Step 1: Understand the Binomial Expansion Formula

The binomial expansion formula is given by the Binomial Theorem, which states that \((x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\), where \(\binom{n}{k}\) is the binomial coefficient. For our problem, \((2a + b)^6\), \(x = 2a\), \(y = b\), and \(n = 6\).

Step 2: Calculate Binomial Coefficients

First, determine the binomial coefficients using \(\binom{6}{k}\) for each term where \(k\) ranges from 0 to 6. Recall that \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).

Step 3: Expand the Expression

Apply the binomial theorem to expand the expression:\[(2a + b)^6 = \sum_{k=0}^{6} \binom{6}{k} (2a)^{6-k} b^k\].Calculate each term:- When \(k=0\): \(\binom{6}{0}(2a)^6b^0 = 1 \cdot 64a^6 \cdot 1 = 64a^6\)- When \(k=1\): \(\binom{6}{1}(2a)^5b^1 = 6 \cdot 32a^5 \cdot b = 192a^5b\)- When \(k=2\): \(\binom{6}{2}(2a)^4b^2 = 15 \cdot 16a^4 \cdot b^2 = 240a^4b^2\)- When \(k=3\): \(\binom{6}{3}(2a)^3b^3 = 20 \cdot 8a^3 \cdot b^3 = 160a^3b^3\)- When \(k=4\): \(\binom{6}{4}(2a)^2b^4 = 15 \cdot 4a^2 \cdot b^4 = 60a^2b^4\)- When \(k=5\): \(\binom{6}{5}(2a)^1b^5 = 6 \cdot 2a \cdot b^5 = 12ab^5\)- When \(k=6\): \(\binom{6}{6}(2a)^0b^6 = 1 \cdot 1 \cdot b^6 = b^6\).

Step 4: Combine All Terms

Combine all the terms gotten from the expansion:\[64a^6 + 192a^5b + 240a^4b^2 + 160a^3b^3 + 60a^2b^4 + 12ab^5 + b^6\].

Key Concepts

Binomial Theorem

The Binomial Theorem is a fundamental concept in algebra used to expand expressions that are raised to a power. The theorem provides a formula that expresses the power of a binomial expression, like \((x + y)^n\), as a sum involving terms of the form \(\binom{n}{k} x^{n-k} y^k\). This formula allows for efficient calculation and expansion of binomial expressions, which would otherwise be cumbersome if done through repeated multiplication.
The binomial theorem is applicable not just for simple expressions involving constants, but also for more complex ones, such as when the terms of the binomial represent algebraic expressions themselves. This theorem is vital when dealing with polynomial expressions that are in factored form and require expansion.

Binomial Coefficients

Binomial coefficients play a crucial role in the application of the Binomial Theorem. These coefficients are represented as \(\binom{n}{k}\), which is read as "n choose k." They determine the weight that each term in the expanded expression will have. The coefficient is calculated using the formula \[\binom{n}{k} = \frac{n!}{k!(n-k)!}\].
Here, \(n!\) (n factorial) is the product of all positive integers up to \(n\), and \(k!\) is the product of all positive integers up to \(k\). These coefficients are symmetric and appear in sequences known as Pascal's Triangle,which beautifully illustrates the recursion inherent in their computation. Binomial coefficients not only help in expansion but also have applications in statistics, combinatorics, and probability.

Polynomial Expansion

A polynomial expansion is the process of breaking down a binomial expression raised to a power into an extended polynomial form. Using the Binomial Theorem, we can expand expressions like \((2a + b)^6\) step-by-step into terms that involve combinations of powers and products.
This technique allows us to view complex algebraic expressions as a series of simpler, managable terms, making it easier to evaluate or manipulate these expressions. Each term in a polynomial expansion follows a distinct pattern of powers and coefficients—dictated by the binomial coefficients and the original terms—in a systematic way. Most importantly, expanding polynomials is foundational for more advanced algebraic concepts, enabling the simplification of equations and the comprehension of polynomial behavior.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operation symbols. They are the building blocks of algebra and are used extensively in describing real-world situations in mathematical form. These expressions can vary from simple, containing a single term, to complex, involving multiple terms connected by various algebraic operations.
An expression like \(2a + b\), when raised to a power, requires expansion, which is where the Binomial Theorem and its related concepts come into play. Understanding and manipulating algebraic expressions through polynomial expansion and the usage of binomial coefficients prepares students for performing algebraic manipulations efficiently and accurately in different mathematical contexts, including problem-solving in calculus, engineering, physics, and economics.