Use the Binomial Theorem to expand the binomial and express the result in simplified form. (2x+1)^3

Verified step by step guidance

1. Identify the binomial expression \((a + b)^n\) where \(a = 2x\), \(b = 1\), and \(n = 3\).
2. Apply the Binomial Theorem: \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\).
3. Calculate each term of the expansion using the formula: \(\binom{3}{k} (2x)^{3-k} (1)^k\) for \(k = 0, 1, 2, 3\).
4. Compute the binomial coefficients: \(\binom{3}{0} = 1\), \(\binom{3}{1} = 3\), \(\binom{3}{2} = 3\), \(\binom{3}{3} = 1\).
5. Combine the terms to form the expanded expression: \(\binom{3}{0} (2x)^3 + \binom{3}{1} (2x)^2 + \binom{3}{2} (2x)^1 + \binom{3}{3} (2x)^0\).

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 simplifies the process of expansion by providing a systematic way to calculate each term.

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 without regard to the order of selection. They can be calculated using the formula C(n, k) = n! / (k!(n-k)!), where '!' denotes factorial. These coefficients play a crucial role in the expansion of binomials, determining the coefficients of each term in the expanded form.

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 with the same variable powers and coefficients. This process is essential for making the final result more manageable and easier to interpret, especially when dealing with polynomials.