Question

Use the Binomial Theorem to expand each binomial. $$(a+b)^{3}$$

   Use the Binomial Theorem to expand each binomial.
$$(a+b)^{3}$$

College Algebra

College Algebra

R. David Gustafson,… 11th Edition

Chapter 8, Problem 25

Instant Answer

Solved by Expert Amy Jiang

07/06/2020

Step 1

Step 1: The Binomial Theorem states that for any numbers a and b, and any natural number n, the expansion of $(a+b)^n$ is given by: $$(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k}$$ where $\binom{n}{k}$ is the binomial coefficient, which is the number of ways Show more…

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Use the Binomial Theorem to expand each binomial. $$(a+b)^{3}$$

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Key Concepts

Binomial Theorem

The Binomial Theorem provides a formula for expanding expressions that are raised to a power, specifically expressions of the form (a+b)^n. It states that such an expression can be written as the sum of terms involving binomial coefficients multiplied by successive powers of the two terms. This theorem is fundamental in algebra and combinatorics for understanding how power sums of binomials can be decomposed into simpler polynomial components.

Binomial Coefficients

Binomial coefficients are the numerical factors that appear in the expansion of a binomial raised to a power. They are calculated using combinations, specifically using the formula 'n choose k', and represent the number of ways to choose k objects from a set of n objects without regard to order. These coefficients play a key role in determining the weight of each term in a binomial expansion.

Pascal's Triangle

Pascal's Triangle is a geometric representation of binomial coefficients arranged in a triangular format. Each row of the triangle corresponds to the coefficients of the expanded form of a binomial raised to a power. This tool simplifies finding the coefficients in binomial expansion and illustrates the symmetric properties and recursive relationships inherent in the binomial coefficients.

Polynomial Expansion

Polynomial expansion involves expressing powers of polynomial expressions as a sum of individual terms with coefficients and variable powers. In the context of the Binomial Theorem, it refers to the process of expanding (a+b)^n into a series of terms such that each term contains a product of a power of a and a power of b, with the exponents summing to n. This concept is essential for simplifying expressions and solving algebraic problems.

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