Find the term indicated in each expansion. (x² + y)²²; the term containing y¹⁴

Verified step by step guidance

1. Identify the general term in the binomial expansion of \((x^{2} + y)^{22}\). The general term is given by the binomial theorem as: \(T_{k+1} = \binom{22}{k} (x^{2})^{22-k} y^{k}\), where \(k\) ranges from 0 to 22.
2. Rewrite the general term to express the powers clearly: \(T_{k+1} = \binom{22}{k} x^{2(22-k)} y^{k}\).
3. Since we want the term containing \(y^{14}\), set the exponent of \(y\) equal to 14, so \(k = 14\).
4. Substitute \(k = 14\) into the general term formula to get the specific term: \(T_{15} = \binom{22}{14} x^{2(22-14)} y^{14}\).
5. Simplify the exponent of \(x\): \$2(22-14) = 2 \times 8 = 16\(, so the term is \)T_{15} = \binom{22}{14} x^{16} y^{14}$.

Key Concepts

Binomial Theorem

The Binomial Theorem provides a formula to expand expressions of the form (a + b)^n into a sum of terms involving binomial coefficients. Each term is given by C(n, k) * a^(n-k) * b^k, where C(n, k) is the binomial coefficient representing combinations.

Binomial Coefficients

Binomial coefficients, denoted as C(n, k) or "n choose k," count the number of ways to choose k elements from n without regard to order. They are calculated using factorials: C(n, k) = n! / (k! (n-k)!). These coefficients determine the coefficients of terms in binomial expansions.

Identifying Specific Terms in a Binomial Expansion

To find a term containing a specific power of a variable in a binomial expansion, set the exponent of that variable equal to the desired power and solve for the term index k. Substitute k back into the general term formula to find the exact term with the required variable power.