Problem 63 - College Algebra

Question

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

Step-by-Step Solution

Answer

The number of terms in a binomial expansion is given by \(n+1\), where 'n' is the power to which the binomial expression is raised.

Step 1: Understand Binomial Expansion

A binomial is an algebraic expression of the sum or the difference of two terms. Binomial expansion refers to expanding an expression that is raised to a power, in its potential series of terms. The power to which the binomials are raised is extremely relevant here.

Step 2: Understand the Binomial Theorem

The Binomial Theorem tells us how to expand a binomial expression that is raised to a power. It is given by: \((a + b)^n = a^n + ^nC_1 * a^{n-1} * b + ^nC_2 * a^{n-2} * b^2 + ^nC_3 * a^{n-3} * b^3 + ... + b^n \) From the above formula, it is visible that the number of terms in the expansion can be found by easily counting the terms.

Step 3: Finding the Number of terms

The number of terms in a binomial expression's expansion is given by \(n+1\), where 'n' is the power to which the binomial expression is raised. For instance, if we are asked to find the number of terms in the expansion of \((x+y)^5\), according to our understanding, the number of terms would be \('n+1'\) =5+1=6. So, the number of terms in the expansion of \((x+y)^5\) will be 6.