In Exercises 9–30, use the Binomial Theorem to expand each binomial and express the result in simplified form. (2x + 1)^4

Verified step by step guidance

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

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. This theorem allows for systematic expansion without the need for repeated multiplication.

Binomial Coefficients

Binomial coefficients, denoted as C(n, k) or 'n choose k', represent the number of ways to choose k elements from a set of n elements. They are calculated using the formula C(n, k) = n! / (k!(n-k)!), where '!' denotes factorial. In the context of the Binomial Theorem, these coefficients determine the weight of each term in the expansion.

Simplification of Expressions

Simplification involves combining like terms and reducing expressions to their most concise form. In the context of binomial expansions, this means collecting terms with the same variable powers and coefficients. This process is essential for presenting the final result clearly and efficiently, making it easier to interpret and use in further calculations.