Problem 56 - College Algebra

Question

How do you determine how many terms there are in a binomial expansion?

Step-by-Step Solution

Answer

To determine the number of terms in a binomial expansion, simply add 1 to the exponent of the binomial. For example, in the binomial expansion of \((a+b)^n\), there are \(n+1\) terms.

Step 1: Understanding Binomial Expansion

In a binomial expansion, each term is formed using the binomial coefficients, whose values are determined by the binomial theorem, and the powers of the terms in the original binomial expression. A binomial expansion is the expansion of a power of a binomial such as \((a+b)^n\).

Step 2: Applying Binomial Theorem

Based on the binomial theorem, the binomial expression \((a+b)^n\) is expanded as \(a^n + \left( ^nC_1 \right) a^{n-1}b + \left( ^nC_2 \right) a^{n-2}b^2 + \ldots + b^n\). Each coefficient in the expansion is a binomial coefficient, as found in the n-th row of Pascal's triangle or computed using the combination formula.

Step 3: Determining the Number of Terms

In an expanded form, the number of terms is one more than the exponent on the binomial expression. This is because the first term is the nth power of the first term in the binomial, the second term is the (n-1)th power of the first term times the first power of the second term, and so on, until the last term which is the nth power of the second term in the binomial.