In Exercises 39–48, find the term indicated in each expansion. (2x + y)^6; third term

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1. Identify the general term in the binomial expansion of \((a + b)^n\), which is given by \(T_{k+1} = \binom{n}{k} a^{n-k} b^k\).
2. For the expression \((2x + y)^6\), set \(a = 2x\), \(b = y\), and \(n = 6\).
3. To find the third term, use \(k = 2\) because the term number is \(k+1\).
4. Substitute \(n = 6\), \(k = 2\), \(a = 2x\), and \(b = y\) into the general term formula: \(T_3 = \binom{6}{2} (2x)^{6-2} y^2\).
5. Calculate the binomial coefficient \(\binom{6}{2}\) and simplify \((2x)^4\) and \(y^2\) to express the third term.

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 in the form C(n, k) * a^(n-k) * b^k, where C(n, k) is the binomial coefficient. This theorem is essential for determining specific terms in the expansion without fully expanding the expression.

Binomial Coefficient

The binomial coefficient C(n, k), also denoted as 'n choose k', represents the number of ways to choose k elements from a set of n elements. It is 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.

Term in a Polynomial Expansion

In a polynomial expansion, a term is a single part of the expression that consists of a coefficient and variables raised to powers. For example, in the expansion of (2x + y)^6, each term corresponds to a specific combination of powers of 2x and y. Identifying a specific term, such as the third term, involves using the Binomial Theorem to find the appropriate coefficients and variable powers.