Explain why the signs alternate in the expansion of (x-y)^9...

Name: Problem 94, Chapter 11: Intermediate Algebra
Uploaded: 2020-04-11T16:01:56-08:00
Duration: 1 min 37 s
Channel: Heather Zimmers
Description: Explain why the signs alternate in the expansion of (x-y)^9

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Key Concepts

Binomial Theorem

The binomial theorem is a fundamental concept used to expand expressions of the form (a + b)^n. It expresses the expansion as a sum of terms where each term consists of a binomial coefficient multiplied by powers of a and b. This theorem provides a systematic way to calculate the coefficients of each term in the expansion.

Binomial Coefficients

Binomial coefficients are the numerical factors in the expansion determined by combinations. They indicate how many ways one can choose a specific number of elements from a set, and in the expansion, they multiply the powers of the individual terms in the binomial expression.

Negative Terms in Binomial Expansion

When expanding a binomial with a negative term, such as (x - y), the negative sign must be considered with each occurrence of the y term. In each term of the expansion, the negative sign is raised to a power corresponding to the exponent of y, which results in alternating positive and negative terms.

Alternating Signs

The alternating sign pattern in the expansion of (x - y)^n arises because the term (-y)^k introduces a factor of (-1)^k. Depending on whether k is even or odd, (-1)^k equals 1 or -1, which makes the signs of the terms alternate in the series.

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