In Exercises 9–30, use the Binomial Theorem to expand each binomial and express the result in simplified form. (2a + b)^6

Verified step by step guidance

1. Identify the binomial expression \((2a + b)^6\) and recognize that the Binomial Theorem can be applied.
2. Recall the Binomial Theorem: \((x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\), where \(\binom{n}{k}\) is the binomial coefficient.
3. Set \(x = 2a\), \(y = b\), and \(n = 6\) in the Binomial Theorem.
4. Calculate each term of the expansion using \(\binom{6}{k} (2a)^{6-k} b^k\) for \(k = 0, 1, 2, \ldots, 6\).
5. Combine all the terms to express the expanded form of \((2a + b)^6\) in a simplified manner.

Key Concepts

Binomial Theorem

The Binomial Theorem provides a formula for expanding expressions of the form (a + b)^n, where n is a non-negative integer. It states that (a + b)^n can be expressed as the sum of terms in the form of C(n, k) * a^(n-k) * b^k, where C(n, k) is the binomial coefficient representing the number of ways to choose k elements from n. This theorem is essential for simplifying and calculating powers of binomials.

Binomial Coefficients

Binomial coefficients, denoted as C(n, k) or 'n choose k', are the coefficients in the expansion of a binomial expression. They can be calculated using the formula C(n, k) = n! / (k!(n-k)!), where '!' denotes factorial. These coefficients determine the number of ways to select k elements from a set of n elements and play a crucial role in the expansion of binomials according to the Binomial Theorem.

Simplification of Expressions

Simplification of expressions involves combining like terms and reducing expressions to their simplest form. In the context of binomial expansion, this means collecting all terms that have the same variables raised to the same powers. This process is important for making the final result more manageable and easier to interpret, especially when dealing with higher powers and multiple variables.