In Exercises 9–30, use the Binomial Theorem to expand each binomial and express the result in simplified form. (5x – 1)^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 the expansion can be expressed as a sum of terms involving binomial coefficients, which are calculated using the formula C(n, k) = n! / (k!(n-k)!), where k ranges from 0 to n. This theorem is essential for simplifying expressions involving powers of binomials.

Binomial Coefficients

Binomial coefficients are the numerical factors that appear in the expansion of a binomial expression according to the Binomial Theorem. They represent the number of ways to choose k elements from a set of n elements and are denoted as C(n, k) or 'n choose k'. These coefficients play a crucial role in determining the coefficients of each term in the expanded form of the 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 and exponent after applying the Binomial Theorem. This process is important for making the final result easier to interpret and use in further calculations.