Problem 62 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the specified \(n\) th term in the expansion of the binomial. \((7 x-2 y)^{15}, n=7\)

Step-by-Step Solution

Answer

The 7th term in the expansion of \( (7x-2y)^{15} \) is \( 320320 \cdot x^9 \cdot y^6 \).

Step 1: Identify the components

From the provided binomial \( (7x-2y)^{15} \), we identify the components as \( a = 7x \), \( b = -2y \), and \( n = 15 \)

Step 2: Apply the Binomial Theorem

By applying the Binomial theorem, the \( k \)th term in the expansion of \( (a+b)^n \) is given by \( T_{k+1} = \binom{n}{k} \cdot a^{n-k} \cdot b^{k} \). We need the 7th term, so we will use \( k = 6\).

Step 3: Substitute and Simplify

Replace the variables in the formula with the given values to get \[ T_{7} = \binom{15}{6} \cdot (7x)^{15-6} \cdot (-2y)^6. \] By simplifying, we get \[ T_{7} = 5005 \cdot (7x)^9 \cdot 64y^6 = 320320 \cdot x^9 \cdot y^6. \]

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful algebraic tool used to expand expressions that are raised to a large power. It reveals the pattern behind the expansion of a binomial expression \((a + b)^n\), allowing us to determine each individual term in the expansion without having to multiply everything out manually.
To apply the theorem, remember the formula for the \(k\)th term:

• \(T_{k+1} = \binom{n}{k} \cdot a^{n-k} \cdot b^k\)

Here, \(\binom{n}{k}\) represents a binomial coefficient, a key element that determines the number of ways to choose \(k\) items from \(n\) total items. This leads us to the next concept, combinatorics.
The Binomial Theorem effectively translates the challenge of performing numerous multiplications into a structured mathematical process, making it easier to find a specific term in the expansion. It also lays the groundwork for deeper concepts in combinatorics and polynomial expansions. Don't forget that each term's exponent in "\(a\)" decreases from \(n\) to zero, while "\(b\)"'s exponent increases from zero to \(n\).

Combinatorics

Combinatorics is a branch of mathematics dealing with countable, discrete structures. In the context of the Binomial Theorem, combinatorics helps us understand how to count the different ways to select terms in the expansion of a binomial.
The binomial coefficient \(\binom{n}{k}\) is essential for this. It represents the number of combinations of \(n\) items taken \(k\) at a time without regard to order.

• Formula: \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\)
• This is also known as "n choose k".

In our binomial expansion exercise, we calculated \(\binom{15}{6}\) to find how many different ways we could select terms for the 7th term of the expansion.
Combinatorics provides the framework for these calculations, enabling us to handle probabilities, permutations, and other mathematical fields. Without the principles of combinatorics, finding specific terms in polynomial expansions would be very challenging.

Polynomial Expansion

Polynomial Expansion is the process of expressing a polynomial as a sum of terms based on its powers. In binomial expansions, each term corresponds to a product of coefficients and variables raised to specific powers.
The binomial theorem specifically deals with polynomial expansion when the polynomial consists of two terms.

• Example in context: The binomial \((7x - 2y)^{15}\) expands into multiple terms, each representing a product of the form \(c \cdot a^p \cdot b^q\).

This is vital for understanding how to "break down" large powers into manageable parts. Each term in an expanded polynomial represents a distinct power of the original variables "\(x\)" and "\(y\)", showing how terms accumulate in an organized fashion.
Through repeated application of the polynomial expansion using the binomial theorem, complex algebraic expressions become more accessible and less intimidating, as the expansion leads to a standard form featuring sums, products, and powers.

Exponents

Exponents, or powers, are a fundamental concept in algebra that helps to express repeated multiplication. Understanding how exponents work is essential in binomial expansions, as it dictates how each variable scales in value.
In a binomial expression \((a + b)^n\), exponents determine the power to which each variable "\(a\)" and "\(b\)" is raised in the expanded terms.

• As we move from one term to the next, the exponent of "\(a\)" decreases, and the exponent of "\(b\)" increases by 1 each time, which maintains the sum \(n\).

In our problem, we found the 7th term which involved powers \(a^9\) and \(b^6\), showing how exponents define the structure of each term.
Grasping how exponents operate within the context of polynomial expansions allows you to predict how each term will appear, making the binomial theorem a powerful technique for managing large exponential expressions. This knowledge applies broadly across mathematics, helping in areas like calculus and solving exponential equations.