Question

Expand $(2 x-y)^{7}$ using the binomial theorem.

   Expand $(2 x-y)^{7}$ using the binomial theorem.

Introductory Combinatorics

Introductory Combinatorics

Richard A. Brualdi 5th Edition

Chapter 5, Problem 5

Instant Answer

Solved by Expert Sriram Soundarrajan

11/01/2020

Step 1

Step 1: Recall the binomial theorem, which states that for any non-negative integer n and any real numbers a and b, we have: $(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$ Show more…

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Expand $(2 x-y)^{7}$ using the binomial theorem.

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Key Concepts

Binomial Theorem

The binomial theorem is a fundamental principle in algebra that provides a formula to expand expressions of the form (a + b)^n into a sum involving terms in which each term is a combination of a raised to a power and b raised to a power, multiplied by a binomial coefficient.

Binomial Coefficient

A binomial coefficient, often represented as C(n, k) or 'n choose k', quantifies the number of ways to choose k elements from a set of n elements. It is a central component in the binomial expansion, determining the weight of each term in the series.

Polynomial Expansion

Polynomial expansion refers to the process of expressing a power of a polynomial, such as a binomial, as a sum of terms. Each term in the expansion has a distinct coefficient, a power of the first term and a power of the second term, thereby transforming the compact expression into an explicit sum of individual terms.

Handling Negative Terms

When a binomial contains a negative term, special attention must be given to the sign of each term in the expansion. The negative sign is raised to the power corresponding to its term in the series, which can result in alternating signs in the expanded expression.

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