Problem 47 - A Graphical Approach to Precalculus with Limits

Question

Find the value of \(n\) for which the coefficients of the fifth and eighth terms in the expansion of \((x+y)^{n}\) are the same.

Step-by-Step Solution

Answer

The value of \(n\) is 11.

Step 1: Write the General Term Formula for Binomial Coefficients

The binomial coefficient for the \(k\)-th term in the expansion of \((x+y)^n\) is given by the formula \( \binom{n}{k-1} \). This expression represents the coefficient of the \(k\)-th term.

Step 2: Set Up the Equation for Coefficients

According to the problem, the coefficients of the fifth and eighth terms are the same. This gives us the equation: \( \binom{n}{4} = \binom{n}{7} \).

Step 3: Use the Property of Binomial Coefficients

There is a property of binomial coefficients that states \( \binom{n}{k} = \binom{n}{n-k} \). Applying this property, we have \( \binom{n}{7} = \binom{n}{n-7} \). Our equation now becomes \( \binom{n}{4} = \binom{n}{n-7} \).

Step 4: Solve for n

This equation implies \(4 = n-7\) or \(7 = n-4\). Solving \(4 = n-7\) gives \(n = 11\). For \(7 = n-4\), we also arrive at \(n = 11\). Hence, \(n = 11\).

Key Concepts

Binomial Coefficient

The binomial coefficient is a key concept in algebra that helps in determining specific term coefficients within polynomial expansions. It is denoted as \( \binom{n}{k} \), read as "n choose k", and calculates the number of ways to choose \( k \) elements from a set of \( n \) elements. The formula for the binomial coefficient is given by:

• \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \)

This mathematical principle is crucial for understanding combinations in various instances of probability and algebraic expressions. In binomial expansion, the coefficients are integral in determining the overall expression for each term. For instance, the fifth term of \((x+y)^n\), is represented by \(\binom{n}{4}(x^{n-4}y^4)\). By mastering the calculation of binomial coefficients, one can conveniently operate with each specific term in expanded polynomial forms.

Expansion of Binomial Expressions

The expansion of binomial expressions involves breaking down expressions like \((x+y)^n\) into a sum of individual terms. Using the Binomial Theorem, such expansions are performed in an efficient and systematic manner. The theorem states that:

• \((x+y)^n = \sum_{k=0}^{n} \binom{n}{k}x^{n-k}y^k\)

This theorem provides a structured approach to expanding expressions without manually multiplying and combining terms.
Each term in the expansion of \((x+y)^n\) comes with a unique binomial coefficient \(\binom{n}{k}\), which perfectly ties back into calculating each term’s contribution with appropriate powers of \(x\) and \(y\). This makes simplifying and solving expressions more manageable, especially when dealing with higher powers.

Combinatorics

Combinatorics is a fascinating field of mathematics chiefly concerned with counting, arranging, and analyzing distinct items. It forms the backbone of dealing with polynomial expansions and binomial coefficients. Specifically, it provides the methodology behind determining potential combinations and permutations in sets, a principle prominently used in the binomial theorem.When dealing with binomial expansions, combinatorics helps us understand the distribution and arrangement of terms and coefficients. It allows one to explore the complexity behind choosing terms and assigning appropriate powers, laying the foundation for the formula

• \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \)

This underpins the efficiency and accuracy in measuring how individual terms are formed and contribute to the final expanded polynomial.

Polynomial Expressions

Polynomial expressions are algebraic expressions that consist of variables and coefficients, involving addition, subtraction, multiplication, and non-negative integer exponents. Understanding polynomial expressions is essential when working with binomial expansions, as these expansions result in a polynomial with various terms.In the context of the binomial theorem, polynomial expressions derived from binomial expansions, like \((x+y)^n\), further illustrate the relationships between terms and coefficients. Solving for coefficients, as seen in the original exercise, converts factored binomial terms into separate polynomial terms with unique coefficients and powers.
This process showcases how polynomial expressions manifest in algebra and how they can be manipulated to explore specific values or coefficients, such as identifying terms where coefficients match.