Problem 12 - Algebra 2 and Trigonometry

Question

In \(12-17,\) write the \(n\) th term of each binomial expansion. $$ (x+y)^{15}, n=3 $$

Step-by-Step Solution

Answer

The 3rd term is \(105x^{13}y^2\).

Step 1: Understand the Binomial Theorem

The Binomial Theorem states that for any positive integer \(n\), \((x + y)^n\) can be expanded as \(\sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^{k}\). This provides a formula for each term in the expansion.

Step 2: Identify Binomial Coefficient and Powers

For the \(n\)-th term of the expansion, where \(n=3\), we use the formula \( \binom{15}{3-1} \cdot x^{15-(3-1)} \cdot y^{3-1} \). The term '3rd' refers to \(k=3-1=2\) in the Binomial Theorem.

Step 3: Calculate the Binomial Coefficient

Calculate \(\binom{15}{2}\), which represents the number of ways to choose 2 elements from 15. This equals \(\frac{15!}{2!(15-2)!} = \frac{15 \times 14}{2 \times 1} = 105\).

Step 4: Calculate the Powers of x and y

For \(x\), the power is \(15-(3-1) = 13\). For \(y\), the power is \(3-1 = 2\).

Step 5: Formulate the 3rd Term

Combine the binomial coefficient and the powers of \(x\) and \(y\) to write the 3rd term as: \(105 \cdot x^{13} \cdot y^2\).

Key Concepts

Binomial Theorem

The binomial theorem is a powerful algebraic formula used to expand expressions raised to a power. It's especially useful for expanding binomials of the form \((x + y)^n\). The theorem provides a systematic way to find each term of the expanded expression without having to multiply manually. For a given power \(n\), the expansion involves a sum of terms, each of them including a binomial coefficient, a power of \(x\), and a power of \(y\).
The formula is:\[\ (x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^{k}\ .\]The binomial theorem is immensely helpful since it allows calculating specific terms directly without having to expand the whole expression. Understanding this mathematical principle is key to solving problems efficiently, especially when dealing with higher powers. Using the theorem, we know exactly how many terms will be in the expansion, which is \(n+1\), and how each term is constructed based on the position \(k\).

Binomial Coefficient

The binomial coefficient, denoted \(\binom{n}{k}\), is a crucial part of the binomial theorem. It represents the number of ways to choose \(k\) elements from a set of \(n\) elements, and it's calculated using factorials:
\[\ \binom{n}{k} = \frac{n!}{k!(n-k)!}.\]In the context of the binomial expansion, this coefficient is multiplied by the powers of variables \(x\) and \(y\).
For example, when expanding \((x + y)^{15}\), and finding the 3rd term (where \(n = 3\)), the binomial coefficient is \(\binom{15}{2}\). Here, it is calculated as:

• \(\frac{15 \times 14}{2 \times 1} = 105\).

These coefficients tell us how many ways the terms \(x^{n-k}\) and \(y^{k}\) can be formed in the expansion, essentially weighing each term depending on their combinations.

Powers of Variables

In a binomial expansion, each term is composed of powers of the variables \(x\) and \(y\). These powers depend on the term's position in the expansion. Understanding how they are derived is crucial.
For a term at a specific position \(k\) in the expansion of \((x + y)^n\):

• \(x\) is raised to the power of \(n-k\).
• \(y\) is raised to the power of \(k\).

So, for the 3rd term in an expansion like \((x + y)^{15}\), the powers are determined as follows:

• The power of \(x\) is \(15 - (3-1) = 13\).
• The power of \(y\) is \(3 - 1 = 2\).

This systematic approach ensures that every term in the expansion follows a predictable pattern, maintaining the balance dictated by the binomial theorem. Recognizing these powers helps solve the problem swiftly and accurately by directly applying the required operations.