In Exercises 9–30, use the Binomial Theorem to expand each binomial and express the result in simplified form. (x+2)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 expanding binomials without the need for repeated multiplication.

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. They are calculated using the formula C(n, k) = n! / (k!(n-k)!), where '!' denotes factorial. In the context of the Binomial Theorem, these coefficients determine the weight of each term in the expansion, reflecting the number of combinations of the variables.

Simplifying Expressions

Simplifying expressions involves combining like terms and reducing expressions to their simplest form. In the context of binomial expansions, this means collecting all terms with the same variable powers and coefficients. For example, after expanding (x + 2)³, one would combine any like terms to present the final result in a clear and concise manner, making it easier to interpret and use in further calculations.