Question

Explain the four patterns apparent in a binomial expansion.

   Explain the four patterns apparent in a binomial expansion.

College Algebra

College Algebra

R. David Gustafson,… 11th Edition

Chapter 8, Problem 70

Instant Answer

Solved by Expert Amy Jiang

07/06/2020

Step 1

Coefficients: In a binomial expansion, the coefficients follow a specific pattern known as Pascal's Triangle. Each row of Pascal's Triangle represents the coefficients of the binomial expansion for a specific power. The coefficients for each term in the expansion Show more…

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Explain the four patterns apparent in a binomial expansion.

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Key Concepts

Symmetry of Terms

Symmetry is an inherent pattern in binomial expansions. Not only do the coefficients mirror each other, but there is also a symmetry in the way the individual terms are structured. For every term a^(n-k) * b^(k) in the expansion, there is a corresponding term a^(k) * b^(n-k) when the order is reversed. This symmetry reflects the combinatorial equivalence of choosing k elements out of n and choosing n-k elements out of n.

Exponent Patterns

Another key pattern in a binomial expansion is that in each term, the sum of the exponents of 'a' and 'b' is equal to n. This means that as the exponent of 'a' decreases from n to 0 across the expansion, the exponent of 'b' increases from 0 to n. This systematic decrease and increase ensure that the total power remains consistent across all terms.

Binomial Theorem

The binomial theorem provides a formula for expanding the expression (a + b)^n into a sum of terms involving binomial coefficients. It states that the expansion is the sum of terms of the form C(n, k)a^(n-k)b^(k), where the binomial coefficient C(n, k) represents the number of ways to choose k objects from a set of n.

Coefficients and Pascal's Triangle

The coefficients that appear in the expansion follow a pattern that corresponds to Pascal's Triangle. In each row of Pascal's Triangle, the numbers represent the binomial coefficients, which increase to a maximum value at the center of the row (for even exponent n) and then symmetrically decrease. This pattern helps in quickly identifying the coefficients needed for each term in the expansion.

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