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

Name: College Algebra
Author: Blitzer
ISBN: 9780136970514

Ch. 8 - Sequences, Induction, and Probability

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Blitzer 8th Edition

Ch. 8 - Sequences, Induction, and Probability

Problem 47

Chapter 9, Problem 47

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

Verified step by step guidance

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.

Rewrite the general term to express the powers clearly: $T_{k+1} = \binom{22}{k} x^{2(22-k)} y^{k}$.

Since we want the term containing $y^{14}$, set the exponent of $y$ equal to 14, so $k = 14$.

Substitute $k = 14$ into the general term formula to get the specific term: $T_{15} = \binom{22}{14} x^{2(22-14)} y^{14}$.

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

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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