In Exercises 39–48, find the term indicated in each expansion. (x − 1)^9; fifth term

Verified step by step guidance

1. Identify the general term in the binomial expansion of \((x - 1)^9\), which is given by the formula: \(T_{k+1} = \binom{n}{k} x^{n-k} (-1)^k\), where \(n = 9\).
2. Determine the value of \(k\) for the fifth term. Since \(k\) starts from 0, the fifth term corresponds to \(k = 4\).
3. Substitute \(n = 9\) and \(k = 4\) into the binomial coefficient formula: \(\binom{9}{4}\).
4. Calculate the expression for the fifth term: \(T_5 = \binom{9}{4} x^{9-4} (-1)^4\).
5. Simplify the expression: \(T_5 = \binom{9}{4} x^5\), since \((-1)^4 = 1\).

Key Concepts

Binomial Theorem

The Binomial Theorem provides a formula for expanding expressions of the form (a + b)^n. It states that the expansion can be expressed as a sum of terms involving binomial coefficients, which are calculated using combinations. This theorem is essential for determining specific terms in polynomial expansions.

Binomial Coefficients

Binomial coefficients are the numerical factors that appear in the expansion of a binomial expression. They are denoted as C(n, k) or 'n choose k', representing the number of ways to choose k elements from a set of n elements. These coefficients are crucial for identifying the specific term in the expansion of (x - 1)^9.

Term Position in Expansion

In a binomial expansion, the position of each term can be determined using the formula for the k-th term, which is given by T(k) = C(n, k-1) * a^(n-k+1) * b^(k-1). For the expression (x - 1)^9, understanding how to calculate the fifth term involves substituting the appropriate values into this formula, ensuring accurate identification of the term's coefficients and variables.