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

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Step 1: Recall the Binomial Theorem, which states that \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\), where \(\binom{n}{k}\) is the binomial coefficient \(\frac{n!}{k!(n-k)!}\).

Step 2: Identify the values of \(a\), \(b\), and \(n\) in the given binomial \((2x + 1)^3\). Here, \(a = 2x\), \(b = 1\), and \(n = 3\).

Step 3: Write the expansion using the Binomial Theorem formula. Substitute \(n = 3\): \((2x + 1)^3 = \binom{3}{0}(2x)^3(1)^0 + \binom{3}{1}(2x)^2(1)^1 + \binom{3}{2}(2x)^1(1)^2 + \binom{3}{3}(2x)^0(1)^3\).

Step 4: Calculate each binomial coefficient \(\binom{3}{k}\) for \(k = 0, 1, 2, 3\). For example, \(\binom{3}{0} = 1\), \(\binom{3}{1} = 3\), \(\binom{3}{2} = 3\), and \(\binom{3}{3} = 1\).

Step 5: Simplify each term in the expansion by evaluating the powers of \(2x\) and \(1\), and multiplying by the corresponding binomial coefficient. Combine all terms to express the result in simplified form.

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주요 개념

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