Find the term in the expansion of (x2 + y2)5 containing x4 as a factor.

Ch. 8 - Sequences, Induction, and Probability

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Blitzer 8th Edition

Ch. 8 - Sequences, Induction, and Probability

문제 82

9장, 문제 82

Find the term in the expansion of (x² + y²)⁵ containing x⁴ as a factor.

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Identify the general term in the binomial expansion of \(\left(x^{2} + y^{2}\right)^5\). The general term is given by \(T_{k+1} = \binom{5}{k} (x^{2})^{5-k} (y^{2})^{k}\), where \(k\) ranges from 0 to 5.

Simplify the powers in the general term: \(T_{k+1} = \binom{5}{k} x^{2(5-k)} y^{2k} = \binom{5}{k} x^{10 - 2k} y^{2k}\).

Since we want the term containing \(x^{4}\), set the exponent of \(x\) equal to 4: \(10 - 2k = 4\).

Solve for \(k\): \(10 - 2k = 4 \implies 2k = 6 \implies k = 3\).

Substitute \(k = 3\) back into the general term to find the specific term: \(T_{4} = \binom{5}{3} x^{10 - 2(3)} y^{2(3)} = \binom{5}{3} x^{4} y^{6}\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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 combination of n items taken k at a time. This theorem is essential for expanding (x² + y²)^5.

Binomial Coefficients

Binomial coefficients, denoted as C(n, k), represent the number of ways to choose k elements from a set of n elements. They appear as coefficients in the binomial expansion and can be calculated using factorials or Pascal’s triangle. Understanding these coefficients helps identify the correct term in the expansion.

Exponent Matching in Terms

To find a term containing a specific factor like x^4, you must match the exponents of variables in each term of the expansion. Since the expansion involves powers of x² and y², the exponents are multiples of 2. Identifying the term where the power of x² equals 4 (i.e., (x²)^2 = x^4) is key to solving the problem.