Problem 58 - Precalculus with Limits

Question

In Exercises 53 - 60, find the coefficient of the term in the expansion of the binomial. Binomial \( \quad \quad \quad \) Term \( \left(3x - 4y\right)^{8} \quad \quad \quad ax^6y^2 \)

Step-by-Step Solution

Answer

The coefficient of the \( ax^6y^2 \) term in the expansion of \( \left(3x - 4y\right)^{8} \) is 11,664.

Step 1: Recognize the Relevant Binomial Term

First, identify the term that you are looking for in the binomial expansion, which is \( ax^6y^2 \). In the general form of the binomial theorem, it corresponds to \((3x)^6 * (-4y)^2\). Now, it is just a matter of calculations.

Step 2: Calculate Coefficient

Calculate the coefficient \(a\). According to our last step, \(a = (3^6) * (-4)^2\). The power in \( (-4)^2 \) signifies the term \( y^2 \). Furthermore, the power in \( 3^6 \) signifies the term \( x^6 \). Therefore, the only task left is to multiply these values together, yielding \(a = 3^6 * (-4)^2 \).

Step 3: Simplify

Simplify the calculation. \(3^6 = 729\), \(-4^2 = 16\). So, \(a = 729 * 16 = 11664\).

Key Concepts

Understanding Binomial Expansion

The binomial expansion describes the method of expanding expressions that are raised to a power and composed of two terms—a binomial. For instance, the expression \( (a+b)^n \) represents a binomial raised to an integer exponent \( n \). To expand such an expression, we use the Binomial Theorem, which provides a formulaic approach. According to this theorem, \( (a+b)^n \) can be expanded as a sum of terms in the form \( C(n, k)a^{n-k}b^k \) where \( C(n, k) \) represents the binomial coefficient and \( k \) ranges from 0 to \( n \).

Understanding this process is crucial, as it appears commonly in algebra, probability, and further in calculus. The ability to expand binomials aids in simplifying complex algebraic expressions and solving polynomial equations more effectively. Let's apply this concept to our original exercise involving the expansion of \( (3x - 4y)^8 \). We specifically look for the term that contains \( x^6y^2 \), which is part of the larger expanded form of our binomial expression.

Coefficient Calculation in Binomial Theorem

Coefficient calculation is a key aspect when working with the binomial theorem. The coefficient of a particular term within a binomial expansion can be obtained using the combination formula \( C(n, k) = \frac{n!}{k!(n-k)!} \), where \( n! \) denotes the factorial of \( n \) and \( k \) is the specific term's index within the expansion.

For example, in the binomial \( (3x - 4y)^8 \) when searching for the coefficient \( a \) of the term \( ax^6y^2 \), we need to determine \( k \). In our case, \( k \) is 2 since the \( y^2 \) term is \( (-4y)^2 \) and \( n-k \) is 6 indicating the exponent of \( x \) is 6. To find \( a \) we then use the formula mentioned above. Following our steps, we established that the coefficient \( a = 3^6 * (-4)^2 \) yielding \( a = 11664 \) after simplification.

Exponents and Powers in Algebra

Exponents and powers are fundamental concepts in algebra, reflecting how many times a number is multiplied by itself. For instance, \( 3^6 \) is \( 3*3*3*3*3*3 \) and \( (-4)^2 \) is \( (-4)*(-4) \). Understanding how to work with exponents is essential when dealing with polynomial expressions, particularly in binomial expansion.

Remember that positive integer exponents denote repeated multiplication, while a negative exponent represents the inverse. Zero as an exponent means the value is 1. When calculating the coefficient of a binomial term, as we did with \( 3^6 * (-4)^2 \) in our binomial expansion, it's vital to calculate the powers separately before multiplication to avoid errors and simplify the solving process effectively.