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 to expand: \((2x + 1)^4\).
2. Recall the Binomial Theorem formula: \(\displaystyle (a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k\), where \(\binom{n}{k}\) is the binomial coefficient.
3. Set \(a = 2x\), \(b = 1\), and \(n = 4\). Write out each term of the expansion using the formula: \(\binom{4}{k} (2x)^{4-k} (1)^k\) for \(k = 0, 1, 2, 3, 4\).
4. Calculate each binomial coefficient \(\binom{4}{k}\) and simplify the powers of \$2x$ and \(1\) in each term.
5. Combine all terms to write the full expanded expression, then simplify by multiplying coefficients and powers of \(x\) where applicable.

Key Concepts

Binomial Theorem

The Binomial Theorem provides a formula to expand expressions raised to a power, such as (a + b)^n. It states that (a + b)^n equals the sum of terms in the form C(n, k) * a^(n-k) * b^k, where C(n, k) are binomial coefficients. This theorem simplifies the expansion process without multiplying the binomial repeatedly.

Binomial Coefficients

Binomial coefficients, denoted as C(n, k) or "n choose k," represent the number of ways to choose k elements from n and appear as coefficients in the binomial expansion. They can be calculated using factorials or found in Pascal's Triangle. These coefficients determine the weight of each term in the expanded expression.

Simplifying Polynomial Expressions

After expanding a binomial using the Binomial Theorem, simplifying involves combining like terms and reducing powers where possible. This step ensures the final expression is in its simplest form, making it easier to interpret or use in further calculations.