Question

Use the Binomial Theorem to find the indicated coefficient or term. The coefficient of \(x^{7}\) in the expansion of \((2 x-1)^{12}\)

Short answer

The coefficient of \( x^7 \) is \ -101376 \.

Step-by-step solution

- Understand the Binomial Theorem

Using the Binomial Theorem, the expansion of \( (a + b)^n \) is given by \[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]. Here, \(a = 2x\), \(b = -1\), and \(n = 12\).

- Identify the General Term

The general term in the expansion is given by \[ T_k = \binom{12}{k} (2x)^{12-k} (-1)^k \].

- Set up the Equation for the Desired Coefficient

We need the coefficient of \(x^7\). Set up the equation \[ (2x)^{12-k} = x^7 \]. This simplifies to \[ 2^{12-k} x^{12-k} = x^7 \].

- Solve for k

From the equation \[ x^{12-k} = x^7 \], it follows that \[ 12 - k = 7 \], or \[ k = 5 \].

- Calculate the Coefficient

Plug \( k = 5 \) into the general term to find the coefficient: \[ T_5 = \binom{12}{5} (2x)^{12-5} (-1)^5 \]. Simplifying, \[ T_5 = \binom{12}{5} (2x)^7 (-1)^5 \].

- Final Calculation

Calculate \( \binom{12}{5} \): \[ \binom{12}{5} = \frac{12!}{5!(12-5)!} = 792 \]. The term then becomes: \[ 792 (2x)^7 (-1)^5 = 792 \times 2^7 x^7 \times (-1) = 792 \times 128 \times (-1) x^7 = -101376 x^7 \].

- Extract the Coefficient

From the final term, the coefficient of \( x^7 \) is \ -101376 \.

Key concepts

Understanding Binomial Expansion

The Binomial Theorem is a powerful tool in algebra that lets you expand expressions of the form \( (a + b)^n \). This theorem states that such an expression can be written as a sum of terms involving binomial coefficients. For any positive integer \( n \), the expansion is:
\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]
where \( \binom{n}{k} \) denotes the binomial coefficient. In this theorem:

• \( a \) and \( b \) are any numbers or algebraic expressions
• \( n \) is a non-negative integer
• The summation runs from \( k = 0 \) to \( k = n \)

In practical terms, this allows you to break down and calculate each term in the expansion one by one. This step-by-step approach simplifies finding specific coefficients or terms within a polynomial.

Understanding Combinatorial Coefficients

A vital part of using the Binomial Theorem involves understanding combinatorial coefficients. These coefficients, expressed as \( \binom{n}{k} \), tell you how many ways you can choose \( k \) elements out of a set of \( n \) elements without regard to the order. The formula for the binomial coefficient is:
\[ \binom{n}{k} = \frac{n!}{k! (n-k)!} \]
where \( n! \) (n factorial) is the product of all positive integers up to \( n \). For example, in the problem we are looking at, we need to calculate \( \binom{12}{5} \). This is computed as:
\[ \binom{12}{5} = \frac{12!}{5!(12-5)!} = 792 \]
This binomial coefficient tells us the number of ways to choose 5 terms out of 12. Understanding this helps us find the coefficient of specific terms within a binomial expansion.

Polynomial Term Identification

To pinpoint a specific term in a binomial expansion, you need to identify how each component in \( (a + b)^n \) contributes to the desired term. Consider the general term of the binomial expansion:
\[ T_k = \binom{n}{k} a^{n-k} b^k \]
In the given problem, the specific term we are interested in is where the power of \( x \) is 7. Given \( (2x - 1)^{12} \), the general term is:

• \( a = 2x \)
• \( b = -1 \)
• \( n = 12 \)

To find the term where the power of \( x \) is 7, we set up the equation:
\[ (2x)^{12-k} (-1)^k = x^7 \]
Solving for \( k \) gives us the value \( k = 5 \). Hence, we substitute \( k = 5 \) back into our general term formula to find the desired term.

Factorial Calculation

Factorial calculations are essential for computing binomial coefficients. The factorial of a non-negative integer \( n \), denoted as \( n! \), is the product of all positive integers up to \( n \):
\[ n! = n \times (n-1) \times ... \times 3 \times 2 \times 1 \]
In our specific binomial expansion problem, to calculate \( \binom{12}{5} \), we must compute:
\[ 12! = 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 479001600 \]
We also need \( 5! \) and \( 7! \):

• \[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]
• \[ 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \]

Plugging these into our binomial coefficient formula, we get:
\[ \binom{12}{5} = \frac{479001600}{120 \times 5040} = 792 \]
Understanding how to compute factorials is crucial for effectively utilizing the Binomial Theorem.