Use the Binomial Theorem to expand each binomial and express the result in simplified form. $(3x+y{)}^3$

Verified step by step guidance

1. Identify the binomial expression to expand: \((3x + y)^3\).
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 \(a = 3x\), \(b = y\), and \(n = 3\). Write out each term of the expansion using the formula: \(\binom{3}{k} (3x)^{3-k} y^k\) for \(k = 0, 1, 2, 3\).
4. Calculate each binomial coefficient: \(\binom{3}{0}\), \(\binom{3}{1}\), \(\binom{3}{2}\), and \(\binom{3}{3}\), and simplify the powers of \$3x\( and \)y$ accordingly.
5. Write the full expanded expression by summing all terms and simplify each term by multiplying coefficients and combining like factors.

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 involving binomial coefficients, powers of a, and powers of b. 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. They appear as coefficients in the expanded form of (a + b)^n and can be found using Pascal's Triangle or the formula C(n, k) = n! / (k!(n-k)!).

Exponent Rules and Simplification

When expanding binomials, powers of each term must be calculated using exponent rules, such as (x^m)(x^n) = x^(m+n). After expansion, like terms should be combined and simplified to express the result in its simplest form, ensuring clarity and correctness.