In Exercises 31–38, write the first three terms in each binomial expansion, expressing the result in simplified form. (y³ − 1)^20

Verified step by step guidance

1. insert step 1: Identify the binomial expression and the power. Here, the binomial is \((y^3 - 1)\) and the power is 20.
2. insert step 2: Use the Binomial Theorem, which states \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\), to expand the expression.
3. insert step 3: Substitute \(a = y^3\), \(b = -1\), and \(n = 20\) into the Binomial Theorem.
4. insert step 4: Calculate the first three terms of the expansion: \(\binom{20}{0} (y^3)^{20} (-1)^0\), \(\binom{20}{1} (y^3)^{19} (-1)^1\), and \(\binom{20}{2} (y^3)^{18} (-1)^2\).
5. insert step 5: Simplify each term by calculating the binomial coefficients and simplifying the powers of \(y^3\) and \(-1\).

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 represent the number of ways to choose elements from a set. Each term in the expansion is given by the formula C(n, k) * a^(n-k) * b^k, where C(n, k) is the binomial coefficient.

Binomial Coefficients

Binomial coefficients, denoted as C(n, k) or 'n choose k', are the coefficients in the expansion of a binomial expression. They can be calculated using the formula C(n, k) = n! / (k!(n-k)!), where '!' denotes factorial. These coefficients determine the number of ways to select k elements from a set of n elements and play a crucial role in the terms of the binomial expansion.

Simplification of Expressions

Simplification of expressions involves reducing complex expressions to their simplest form, making them easier to work with. In the context of binomial expansions, this means combining like terms and reducing any coefficients or variables where possible. For example, in the expansion of (y³ - 1)^20, simplifying the terms will help in clearly identifying the first three terms of the expansion.