Problem 72 - Algebra 2

Question

Which term in the expansion of \((2 a-3 b)^{6}\) has coefficient 2160\(?\) \(\begin{array}{ll}{\text { F. second term }} & {\text { G. third term }} \\\ {\text { H. fourth term }} & {\text { J. fifth term }}\end{array}\)

Step-by-Step Solution

Answer

The term in the expansion of \( (2 a - 3 b) ^ 6 \) with the coefficient 2160 is the 5th term, represented by option J.

Step 1: Identify each term in the binomial expansion

The binomial expansion theorem tells us the general formula for any term in the expansion is \(T _(r+1) = \binom{n}{r} * (2a)^{n-r} * (-3b)^r \). Each term of the binomial expansion takes this form.

Step 2: Determine the coefficient of each term

The coefficient of each term is given by \( \binom{n}{r} * 2 ^ {n-r} * (-3) ^ r \). In our problem, n = 6 (since we are expanding \( (2 a - 3 b) ^ 6 \)). We know that this coefficient should be 2160. We can therefore set up an equation as follows: 2160 = \( \binom{6}{r} * 2 ^ {6-r} * (-3) ^ r \)

Step 3: Solve the equation

Solving this equation for r lets us know which term has a coefficient of 2160. When you solve, you'll find r = 4. This represents the 5th term in the expansion (since we start counting at zero).

Step 4: Give the correct answer

In the given options, the fifth term is represented by option J.

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful tool in algebra. It provides a way to expand expressions of the form \((x + y)^n\). This theorem lets us write these expansions as a sum of simpler terms. Each term in the expansion is composed of binomial coefficients, powers of the first term, and powers of the second term. The general form for any single term in a binomial expansion is given by:

• \(T_{r+1} = \binom{n}{r} \cdot x^{n-r} \cdot y^r\)

where \(T_{r+1}\) is the \((r+1)\)-th term, \(\binom{n}{r}\) is the binomial coefficient, and \(x\) and \(y\) are the terms of the binomial. This formula allows us to compute each term individually, using different values of \(r\).
The usefulness of the binomial theorem lies in its ability to simplify the process of expanding large expressions, which would otherwise be cumbersome and time-consuming. Understanding the binomial theorem is crucial for solving problems related to polynomial expansions.

Binomial Coefficients

Binomial coefficients play a key role in the binomial expansion. These coefficients are represented as \(\binom{n}{r}\), also known as "n choose r". They determine how much of each term contributes to the expansion.
The formula for calculating binomial coefficients is:

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

where \(n!\) is the factorial of \(n\), and it represents the product of all positive integers up to \(n\). The factorial is a fundamental concept in combinations and permutations. Understanding how binomial coefficients work is essential to figuring out the weight of each term in the expansion.
In our given problem, determining the correct binomial coefficient was crucial to finding the term with a coefficient of 2160. Once identified, these coefficients make quite straightforward the otherwise complex task of achieving polynomial expressions through expansions.

Polynomial Expansion

Polynomial expansion is the process of expressing a binomial raised to a power as a polynomial. Expanding a binomial means turning it into a sum of terms, each with different powers of the variables involved, usually \(x\) and \(y\).
While the binomial theorem provides the formula, understanding polynomial expansion requires practicing the application of the theorem to reach the final expression. As each term is calculated, powers of components, as well as their coefficients, are explicitly determined.

• Each term is carefully constructed by deciding powers based on the binomial theorem.
• The expansion reveals all possible combinations of powers that add up to the degree \(n\) of the polynomial.

This method was used to find the term with a coefficient of 2160 in our problem. By expanding \((2a - 3b)^6\), we're transforming the binomial into a detailed polynomial structure. Polynomial expansion serves many practical mathematical applications, such as calculating probabilities and finding roots, alongside solving algebraic equations.