Problem 40 - College Algebra

Question

In Exercises 39-48, find the term indicated in each expansion. $$(x+2 y)^{6} ; \text { third term }$$

Step-by-Step Solution

Answer

The third term in the expansion of \((x+2 y)^{6}\) is \(60x^{4}y^{2}\)

Step 1: Understand the Binomial Theorem

The Binomial theorem is given as \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k}\) where \(n\) is a positive integer and \(\binom{n}{k}\) is a binomial coefficient that counts the number of ways to choose k elements from n elements.

Step 2: Calculate the coefficients of each term

The Binomial coefficient is given as: \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\). Here, \(n=6\). We need to find the third term, that is, \(k=2\). So, calculate the binomial coefficient of the third term first: \(\binom{6}{2} = \frac{6!}{2!(6-2)!} = 15\).

Step 3: Determine the Third Term

According to the Binomial theorem, the third term is given by the formula: \(\binom{n}{k} a^{n-k} b^{k}\). By substituting \(n=6\), \(a=x\), \(b=2y\), and \(k=2\) into this, the third term is: \(\binom{6}{2} x^{6-2} (2y)^{2} = 15 * x^{4} * 4y^{2} = 60x^{4}y^{2}\).

Key Concepts

Binomial Expansion

The binomial expansion deals with breaking down expressions that are raised to a power. When you see something like \((a + b)^n\), the Binomial Theorem lets us expand this expression into a series of terms without having to manually multiply \((a + b)\) repeatedly. With the theorem, we can write:\[(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k}\]
In simpler terms, the theorem provides a formula to find each term of the expansion one at a time. Each term of the expansion is a product of a binomial coefficient, a power of \(a\), and a power of \(b\). The expansion results in a polynomial where the sum of the powers of \(a\) and \(b\) in each term always adds up to \(n\).
This is particularly useful when dealing with large exponents, as it systematically breaks down the binomial into a sum using a predictable pattern.

Binomial Coefficients

Binomial coefficients are a key component of the binomial expansion. They tell us how many ways we can pick a certain number of items from a larger set. In the context of the Binomial Theorem, they appear in each term of the expansion and are denoted as \(\binom{n}{k}\). The formula for calculating binomial coefficients is:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]
Here, \(n!\) (read as "n factorial") means the product of all positive integers up to \(n\), and \(k!\) is the factorial of \(k\). The coefficient essentially counts the different ways we can arrange the terms in each part of the expansion.
For example, if we want the coefficient for the third term in the expansion of \((x + 2y)^6\), we calculate \(\binom{6}{2}\), resulting in a coefficient of 15. It gives us the primary multiplying factor for the corresponding term.

Polynomial Expressions

Polynomial expressions are sums of terms that involve variables raised to whole-number exponents. A polynomial is made up of several entities, each called a term, where each term consists of a coefficient, a variable like \(x\) or \(y\), and an exponent.
Consider the expression obtained from binomial expansion. For instance, from the expansion of \((x + 2y)^6\), the polynomial expression for the third term is \(60x^4y^2\). This term combines both coefficients (60, in this case) and variable powers to describe how items like \(x\) and \(y\) appear in this portion of the overall expression.
Understanding polynomial expressions means recognizing how these terms interact and combine to form the complete expression. In binomial expansions, each term follows a pattern determined by the binomial theorem and coefficients, providing clear guidance on how to construct each part of the resulting polynomial.