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Ch. 8 - Sequences, Induction, and Probability
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 8 - Sequences, Induction, and ProbabilityProblem 47
Chapter 9, Problem 47

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

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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.
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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.
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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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Related Practice
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