In Exercises 9–30, use the Binomial Theorem to expand each binomial and express the result in simplified form. (3x + y)^3

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 determining the coefficients of each term in the expansion of a binomial expression.

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 terms with the same variable powers and coefficients. This process is essential for presenting the final result clearly and concisely, making it easier to interpret and use in further calculations.