Problem 64 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the coefficient \(a\) of the given term in the expansion of the binomial. Binomial \((x+4)^{12}\) Term \(a x^{4}\)

Step-by-Step Solution

Answer

The coefficient of \(x^{4}\) in the expansion of \((x+4)^{12}\) is 7920.

Step 1: Identify the Form

Firstly, identify the general form of any term in the binomial expansion using the binomial theorem, which is \(\binom{n}{k} a^{n-k} b^{k}\)

Step 2: Substitute the variables

In the expansion of \((x+4)^{12}\), \(a=x\), \(b=4\), and \(n=12\). In the term we want, \(a x^{4}\), \(k=4\) because the power of \(x\) is 4. Now, we substitute these variables into the binomial theorem, obtaining \(\binom{12}{4} x^{12-4} 4^{4}\)

Step 3: Calculate the Coefficient

Substituting the values simplifies the expression to be \(\binom{12}{4} x^{8} 4^{4}\), which simplifies to 495*16 = 7920. So the coefficient of \(x^{4}\) in the expansion \((x+4)^{12}\) is 7920

Key Concepts

Binomial Theorem

The Binomial Theorem is a fundamental principle in algebra that provides a way to expand expressions raised to a power. For any positive integer \( n \), the expansion of \( (a+b)^n \) is given by:

• \( \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k} \)

Here, \( \binom{n}{k} \) represents the binomial coefficient, which dictates the number of ways to choose \( k \) elements from \( n \). This theorem is exceptionally useful when dealing with polynomials and allows for easy calculation of individual terms within the expanded form.
Adding simplicity to binomial expansions, this method is widely applied in higher-level mathematics, including calculus.

Coefficient Calculation

Coefficient calculation within a binomial expansion involves finding the numerical factor of a specific term. For a term in the expansion of \( (x+4)^{12} \) like \( a x^4 \), the process utilizes the binomial coefficient and the powers of the variable and constant.

• Identify \( k \) so that the term involves \( x^k \).
• Substitute into the formula: \( \binom{12}{4} x^{12-4} \cdot 4^4 \).
• Perform calculations, like \( \binom{12}{4} = 495 \) and \( 4^4 = 256 \).
• Multiply to find the coefficient: \( 495 \times 256 = 7920 \).

This process ensures that each term is precisely calculated, capturing its full value within the polynomial expression.

Combinatorics

Combinatorics involves counting, arranging, and finding patterns but is particularly useful in binomial expansions through binomial coefficients. A binomial coefficient \( \binom{n}{k} \) is derived from combinatorial logic, representing the number of ways to choose \( k \) elements from \( n \) without regard to order.

• Calculate \( \binom{n}{k} \) using: \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \).
• In the exercise, \( \binom{12}{4} \) is calculated as \( \frac{12!}{4! \, 8!} \), resulting in 495 ways.

Understanding combinatorial frameworks provides insights into how algebraic structures are formed and expanded.

Polynomial Expansions

Polynomial expansions transform expressions into an expanded sum of terms. In the case of binomials, this refers to the complete spread of terms according to the Binomial Theorem. Each step of expanding \((x+4)^{12}\) involves:

• Determining each term’s contribution based on powers and coefficients.
• Utilizing the calculated binomial coefficients and powers to write out terms such as \( \binom{12}{4} x^8 \cdot 256 \).
• Summing these terms creates the expanded polynomial.

This approach not only simplifies complex expressions but also prepares them for further analysis or applications, such as solving equations or analyzing functions.